In negative-index materials (NIMs), the self-steepening (SS) effect is proven to be anomalous in two aspects: First, it can be either positive or negative, with the zero SS point determined by the size of split-ring resonator circuit elements. Second, the negative SS parameter can have a very large value compared to an ordinary positive-index material. We present a theoretical investigation on modulation instability (MI) to identify the role of the anomalous SS effect in NIM. We find that the first anomaly of SS doesn’t influence MI, yet the controllable zero SS point can be used to manipulate MI, and thus manipulate the generation of solitons. The second anomaly, however, leads to significant changes in the MI condition and property, compared with the case of an ordinary positive-index material. Numerical simulations confirm the theoretical results and show that negative SS moves the center of generated pulse toward the leading side, and shifts a part of energy of the generated pulse toward the red side, opposite to the case of positive SS.
© 2006 Optical Society of America
Modulation instability (MI) is one of the most fundamental processes in nonlinear wave systems in nature. In a nonlinear dispersive material, it stems from the interaction between the nonlinear effect and group velocity dispersion (GVD), and in general, is characterized by instability experienced by a continuous wave (cw) when it propagates together with a weak noise . Because of MI, small amplitude perturbations that originate from noise on top of a homogeneous wave front grow rapidly under the combined effects of nonlinearity and dispersion. As a result, a quasi-cw pulse breaks up into a train of short pulses. It is important to note that MI is an issue closely related to the existence of both bright and dark solitons; it can be considered as a precursor to soliton formation.
Over the years MI in nonlinear dispersive materials such as optical fibers, has been systematically investigated in connection with numerous nonlinear processes . All the investigations, however, are for the ordinary positive-index materials. Various higher-order effects, such as higher-order dispersions, self-steepening and time-delayed Raman effects, have been found to strongly influence MI. It has been found that third-order dispersion (TOD) contributes none to instability[2, 3], while fourth-order dispersion (FOD) plays a major role in the critical MI frequency and may lead to additional instability regions [4, 5]. SS narrows the MI frequency region [3, 5, 6] and the time-delayed Raman nonlinearity alters MI fundamentally .
In this paper, we investigate MI in negative-index material (NIM) with Kerr nonlinearity. NIMs’ most impressive property is their ability to refract light in the opposite way with respect to what an ordinary material does [7, 8, 9, 10]. NIMs were realized primarily in the microwave range. Nowadays, NIMs in the near IR and optical range have also been experimentally demonstrated [11, 12, 13]. Additionally, nonlinear NIMs can also be created [14, 15, 16]. For example, Zharov et al.  demonstrated that a two dimensional periodic structure created by arrays of wires and split-ring resonators embedded into a nonlinear dielectric takes on a Kerr-type dielectric permitivity. Very recently, authors have investigated the nonlinear propagation of ultrashort pulse in NIM [17, 18, 19, 20]. In Ref. Lazarides and Tsironis derived a system of coupled nonlinear Schrödinger equations (NLSEs) for the envelopes of the propagating electric and magnetic fields in an isotropic, homogeneous, quasi-one-dimensional NIM under the conditions of nonlinear polarization and nonlinear magnetization. Based on the coupled NLSEs, Kourakis and Shukla have investigated the nonlinear stability of electromagnetic waves in NIM, and obtained the modulational stability profile of the coupled plane-wave solutions . Scalora et al.  investigated the propagation of pulses at least a few tens of optical cycles in duration in NIM with only a nonlinear polarization. They also derived a system of coupled nonlinear equations. By eliminating the magnetic field, the system leads to a new generalized NLSE in which the linear and nonlinear coefficients can be tailored through the linear properties of the medium to attain any combination of signs unachievable in ordinary matter, showing significant potential to realize a wide class of solitary waves. The aim of this paper is to disclose the new MI properties in NIM. Especial attention is payed to the role played by the anomalous SS effect in NIMs. First, we derive a propagation model for ultrashort pulse in NIM charaterized by a Drude model of both ε and μ,under the condition of a nonlinear polarization, and compare it to that in an ordinary material. Unlike Refs.[18, 20], we eliminate the magnetic field at the very start. This makes it possible for us to derive the propagation equation by the same way as in ordinary materials. Compared to Ref., we obtain a more general NLSE suitable for few-cycle pulse propagation. By using the same approximations, our equation is reduced to that of Ref.. Second, we present a linear stability analysis to identify the role of the anomalous SS effect in MI. Third, we give a numerical simulation to confirm the obtained results and to further demonstrate that NIMs provide new choice for soliton generation.
2. Modelling the pulse propagation in NIM
We assume that the pulse is propagating in uniform, bulk meterials under the condition of a nonlinear polarization . For simplicity we consider one longitudinal spatial coordinate and time. We thus start with the following wave equation
The electric field E propagates along the z direction. Both E and the nonlinear polarization Pnl are assumed to be polarized parallel to the x axis. The dielectric susceptibility ε and magnetic permeability μ are dispersive in a NIM, otherwise the energy density could be negative . Their frequency dispersion can be described by a lossy Drude model :
where ω is frequency, ωpe and ωpm are the respective electric and magnetic plasma frequencies, γe and γm are the respective electric and magnetic loss terms, which are very small and are neglected in the following analysis for simplicity, and ε 0 and μ 0 are the respective vacuum susceptibility and magnetic permeability.
We can transform Eq. (1) into frequency space in order to expand ε(ω) and μ(ω) in powers of ω, thus enabling us to treat the material parameters as a power series whichwe can truncate to an approriate order Howeverfor simplicity it is better to expand ωε(ω) and ωμ(ω) abouta suitable ω 0 instead,
We can now write the frequency space version of Eq. (1) as
We introduce an envelope and carrier form for the field in the usual way, E(z,t) = (1/2)A(z,t)exp(ik 0 z - iω 0 t) + c.c, where k 0 = n(ω 0)ω 0/c is wave number in medium at the carrier frequency ω 0, n(ω 0) is the refractive index of medium at ω 0, and assume that, Pnl (z,t) = ε 0χ(3) ∣E(z,t)∣2 E(z,t), which characterizesaKerrnonlnearity. Withtheseenvelope-carrier substitutions, taking the inverse Fourier transform of the obtained equation, we have
where V = 2k 0/(Θ0Ξ1 + Θ1Ξ0) is the group velocity of the pulse, , ϒ m =m!ω 0 ε 0χ(3)Ξ m /(2k 0). Introducing co-moving variables, τ = t - z/V, ξ = z, Eq. (6) is transformed to the following generalized NLSE
is the GVD which can be easily proven to satisfy the definition of GVD, β 2 = ∂ 2 k/∂ ω 2∣ω0.
In deriving Eq. (7), we have not made any further approximations. Thus it is suitable for the propagation of ultrashort pulse with few optical cycles in NIMs. It is formally similar to the models for few-cycle pulse propagation in ordinary positive-index materials . In addition, by using the same approximations,Eq. (7) is reduced to Eq. (12) of . The mostnotable characteristic of the propagation equation for NIM is the anomalous self-steepening (SS) parameter. To exclusively demonstrate the role of SS in MI in NIM, we keep the dispersion coefficients to second order and neglect the higher order derivatives with respect to the nonlnearity, and use the same estimations as : ∂ 2 A/∂ ξ 2 = iϒ0 ∂(∣A∣2 A)/∂ξ, ∂ 2 A/∂τ∂ξ = iϒ0 ∂(∣A∣2 A)/∂τ. Thus we obtainthe following NLSE
where Cnl = ϒ0 and Cs = 1 + ω 0ϒ1/ϒ0 - ω 0(k 0 V) are the nonlinear and SS coefficients, respectively. For lossless Drude model, the expressions for β 2, Cnl and Cs are
Equation (9) is formally the same as that for nonlinear pulse propagationin ordinary positive-index materials. The noticeable anomalyin Eq.(9) is the SS parameter Cs . Inits peer in ordnary positi ve-index materials, Cs = 1. To disclose the anomalous properties of the SS parameter, we plot the variation of n, Cs , and β 2 with ω/ωpe for different values ofthe ratio ωpm /ωpe for γe = γm = 0 in Fig. 1. We see that, first, like GVD [10, 21], SS can also be engineered by choosing the size of spit-ring resonator circuit elements. Second, the SS parameter can be negative under the condition of ωpm /ωpe < 1, with the zero SS point located at
Third, in the positive SS region, the normalized SS parameteris less than 1; while in the negative SS region, its value can be fa r greater than 1, especially in the normal GVD regime.
The possibility of controlling the zero GVD point of a NIM can be exploited to obtain dspersion-free propagationin spectral regions otherwise inaccessible using ordinary positive-index materials [10, 21]. Smilarly, by controlling the SS in NIM, we believe thatwe can obtain some new properties of the nonlinear pulse propagation, such as solitons and MI.
3. Role of the anomalous self-steepening in MI in NIM
In this Section we identify the role of the anomalous self-steepening in MI in NIM. For simplicity, we first introduce the normalized variables T = τ/τp , Z = ξ/ld , and U = A/A 0, where τp is the width ofthe input pulse, ld = /∣β 2∣ the dispersion length, lnl = 1/(∣Cnl ∣) the nonlinear length, and A 0 the amplitude ofthe input field,to transform Eq. (9) into the normalized form:
where N = ld /lnl , S = Cs /(τp ω 0) is the normalized self-steepening coefficient, δ = ±1 stands for normal and anomalous GVD respectively, and ϑ = ±1 for focusing and defocusing nonlin-earity respectively.
We then study MI in NIM based on Eq. (14). MIs in ordinary materials based on the similar nonlinear Schrödinger equation has been well understood [1, 3]. By a standard linear stability analysis,we can obtainthe MI growthrate
where ϖ is frequency for perturbation, b = , U 0 is the normalized initial amplitude ofthe wave. Eq. (15) shows that MI in NIM appears for focusing nonlinearity (ϑ = 1) and anomalous dispersion (ϑ = -1), or defocusing nonlnearity (δ = -1) and normaldispersion (δ = 1), as in ordinary materials [1, 3]. The fact that negative refraction doesn’t change the conditions for MI to appear is due to that although negative refration alters the sign of wavenumber k 0, it doesn’t alter the sign of nonlnearity coefficient determined by the combination of signs of μ, nand χ(3), as Eq.(11) showns. For NIM, μ and n are simultaneously negative, and thus the sign of nonlinearity coefficient is only determined by χ(3), as in ordinary materials From Eq.(15), we can obtain the critical frequency for MI to occur, and the fastest growing frequency for occurring MI
The role of the anomalous SS effect in MI in NIMs is clearly seen in Eqs.(15) and (16). Firstly, although the SS parameter can be either positive or negative, MI is irrespective of the sign of SS parameter. Secondly, for some frequencies, the value of SS parameter is very large, as Fig. 1 shows, possibly enough to make the MI growth rate and the critical frequency be imaginary, meaning MI can’t occur in NIM. Figure 2(a) is the contour plot of MI growth rate versus ϖ and SS parameter; and Fig.2(b) is the variation of the critical frequency and the fastest growing frequency with SS parameter. We see that the SS effect leads to the reduction of the gain and movement of the positions of the critical frequency and the fastest growing frequency to lower frequencies. As the SS parameter increases further to exceed a critical value, MI can’t appear. As stated before, SS effect can be engineered, thus the MI can be manipulated. These results illustrate not only the unusual nonlinear effects that can be seen in NIMs, but also the new ways of manipulating solitons.
4. Numerical simulations
To further disclose the role of SS in MI and to confirm the obtained properties of MI in NIMs we numerically solve Eq.(14) using the standard split-step Fourier method for ωpm /ωpe = 0.8 in the case of focusing nonlinearity and anomalous dispersion. In this case, the SS parameter can vary from 0.867 to -1.5, with the zero SS point located at ω/ωpe = 0.632, as Fig. 1(b) shows. The initial field distribution is a cosinoidally modulated plane wave,
where U 0 is the initial amplitude of background wave which is set to be 1 in the following simulations, u 0 = 0.05 is the initial amplitude of modulation wave. In a laboratory experiment, noise in the system provides the seed from which instability frequencies begin to develop. However, for the numerical simulations, it is necessary to explicitly provide an initial seed at the frequency ϖ to stimulate a response (induced MI). In the following simulations,ϖ is chosen such that it is the fastest growing frequency in the case of S = 0.
Figure 3 shows the temporal and spectral distribution of the field intensity in the cases of zero SS (S = 0), positive SS (S = 0.2) and negative SS (S = -0.2), at different propagation distances into the NIM. It is shown that in all the three cases, the cosinoidally modulated plane wave evolves into a train of pulses with much higher amplitude than the initial modulation. The period of the train equals 2π/ϖ. In addition, we can find, (i) compared to the zero SS case, both positive and negative SS slow down the growth rate of the modulation, in accordance with the prediction of the linear stability theory, and (ii) as the propagation distance increases, the negative SS moves the center of modulated wave or generated pulse toward the leading side, and shifts a part of energy of the generated pulse toward the red side, opposite to the case of positive SS. As is well known, MI is closely related to the phenomenon of soliton. The important difference between negative and positive SS effects in MI also manifests in solitons, as Fig. 4 shows. It clearly shows that, as the propagation distance increases, the second order soliton moves to the leading sides due to the role of the negative SS effect, opposite to the case of positive SS.
We have derived a NLSE for ultrashort pulse propagation in NIM with Kerr nonlinearity. The derived equation is formally similar to the case of ordinary positive-index materials, but has an anomalous SS parameter: it can be either positive or negative, and can have a very large negative value. Based on the obtained NLSE, we have analyzed MI in NIM using the standard linear stability analysis. It is shown that the conditions for MI to occur in NIM are the same as in ordinary materials if SS effect are neglected. The sameness is due to the fact that although negative refraction reverses the phase velocity and thus the sign of the wavenumber, it doesn’t alter the signs of dispersion and nonlinearity. The most notable property of MI in NIM is that it can be manipulated by engineering SS effect by choosing the size of split-ring resonator circuit elements. Numerical simulations confirms the linear analysis results and show that negative SS moves the center of generated pulse toward the leading side, and shifts a part of energy of the generated pulse toward the red side, opposite to the case of positive SS. The important difference between negative and positive SS effects in MI suggests that we can manipulate solitons in a new way.
About a month later after we had submitted this manuscript, we were reminded that Marklund etal. also studied the MI in metamaterials .
This work is partially supported by the National Natural Science Foundation of China (Grant Nos. 10576012 and 60538010) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20040532005).
References and links
1. G. P. Agrawal. Nonlinear Fiber Optics, 3nd edn. (San Diego, Academic, 2001).
2. V. A. Vysloukh and N. A. Sukhotskova, “Influence of third-order dispersion on the generation of a train of picosecond pulses in fiber waveguides due to self-modulation instability,” Sov. J. Quantum Electron. 17, 1509 (1987). [CrossRef]
3. M. J. Potosek, “Modulation instability in an extended nonlinear Schrödinger equation,” Opt. Lett. 12, 921 (1987). [CrossRef]
5. F. Kh. Abdullaev, S. A. Darmanyan, S. Bsichoff, P. L. Christiansen, and M. P. Sørensen, “Modulational instability in optical fibers near the zero dispersion point,” Opt. Commun. 108, 60 (1994). [CrossRef]
6. F. Kh. Abdullaev, S. A. Darmanyan, S. Bischoff, and M. P. Sørensen, “Modulational instability of electromagnetic waves in media with varying nonlinearity,” J. Opt. Soc. Am. B 14, 27 (1997). [CrossRef]
07. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of eand μ,” Sov. Phys. Usp. 10, 509 (1968). [CrossRef]
09. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite Medium with zimultane-ously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184 (2000). [CrossRef] [PubMed]
11. A. Berrier, M. Mulot, M. Swillo, M. Qiu, L. Thylén, A. Talneau, and S. Anand, “Negative Refraction at Infrared-Wavelengths in a Two-Dimensional Photonic Crystal,” Phys. Rev. Lett. 93, 73902 (2004). [CrossRef]
12. E. Schonbrun, M. Tinker, W. Park, and J. -B. Lee, “Negative refraction in a Si-polymer photonic Crystal membrane,” IEEE Photonics Technol. Lett. 17, 1196 (2005). [CrossRef]
13. V. M. Shalaev, W. Cai, U. K. Chettiar, H. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett. 30, 3356 (2005). [CrossRef]
15. S. O’Brien, D. McPeake, S. A. Ramakrishna, and J. B. Pendry, “Near-infrared photonic band gaps and nonlinear effects in negative magnetic metamaterials,” Phys. Rev. B 69, 241101 (2004). [CrossRef]
16. M. Lapine, M. Gorkunov, and K. H. Ringhofer, “Nonlinearity of a metamaterial arising from diode insertions into resonant conductive elements,” Phys. Rev. E 67, 065601 (2003). [CrossRef]
17. V. M. Agranovich, Y. R. Shen, R. H. Baughman, and A. A. Zakhidov, “Linear and nonlinear wave propagation in negative refraction metamaterials,” Phys. Rev. B 69, 165112 (2004). [CrossRef]
18. N. Lazarides and G. P. Tsironis, “Coupled nonlinear Schröinger field equations for electromagnetic wave propagation in nonlinear left-handed materials,” Phys. Rev. E 71, 036614 (2005). [CrossRef]
19. I. Kourakis and P. K. Shukla, “Nonlinear propagation of electromagnetic waves in negative-refraction-index composite materials,” Phys. Rev. E 72, 016626 (2005). [CrossRef]
20. M. Scalora, M. S. Syrchin, N. Akozbek, E. Y. Poliakov, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, “Generalized nonlinear Schrödinger equation for dispersive susceptibility and permeability: application to negative index materials,” Phys. Rev. Lett. 95, 013902 (2005). [CrossRef] [PubMed]
21. G. D’Aguanno, N. Akozbek, N. Mattiucci, M. Scalora, M. J. Bloemer, and A. M. Zheltikov, “Dispersion-free pulse propagation in a negative-index material,” Opt. Lett. 30, 1998 (2005). [CrossRef] [PubMed]
22. T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett. 78, 3282 (1997). [CrossRef]
23. M. Marklund, P. K. Shukla, and L. Stenflo, “Ultra-short solitons and kinetic effects in nonlinear metamaterials,” arXiv: nlin./060162.