Nonlinear Wave Propagation in Negative Index

Nikolaos L. Tsitsas1 and Dimitri J. Frantzeskakis2 1 School of Applied Mathematical and Physical Sciences, National Technical University of Athens, Zografos, Athens 15773, Greece 2Department of Physics, University of Athens, Panepistimiopolis, Zografos, Athens 15784, Greece [email protected]

Abstract— Wave propagation in nonlinear negative index meta• In this work, we start from Maxwell's equations, and use the materials is investigated by directly implementing the reductive reductive perturbation method to derive a second-order and a perturbation method to Faraday's and Ampere's laws. In this third-order nonlinear Schrodinger (NLS) equation, describing way, we derive a second-order and a third-order nonlinear Schrodinger equation, describing solitons of moderate and ultra- ultra-short solitons in nonlinear left-handed metamaterials. short pulse widths, respectively. We find necessary conditions and Then, we find necessary conditions and derive exact bright derive exact bright and dark soliton solutions of these equations and dark soliton solutions of these equations for the electric for the electric and magnetic field envelopes. Directions of and magnetic field envelopes. future work towards the modelling of wave propagation in more More precisely, we present a systematic derivation of NLS complicated types of nonlinear negative index metamaterials (e.g., chiral metamaterials) are pointed out. and higher-order NLS (HNLS) equations for the electromag­ netic field envelopes, as well as ultra-short solitons for left- handed metamaterials. In particular, we use the reductive per­ I. INTRODUCTION turbation method to derive from Faraday's and Ampere's laws Metamaterials possessing negative , due to a hierarchy of equations. Using such an approach, i.e., directly simultaneous negative e and permeability /i, have analyzing Maxwell's equations, rather than coupled wave become a subject of intense research activity, since they equations for the envelopes, we show are characterized by remarkable electromagnetic properties, that the envelope is proportional to the magnetic including: negative refraction for interface (reversal field one (their ratio being the linear wave-impedance). Thus, of Snell's law), backward wave propagation, reversal of the for each of the electromagnetic wave components, we derive Doppler shift and Cherenkov effect, collecting lens behavior a single NLS (for moderate pulse widths) or a single HNLS forming 3-D images, perfect lens performance by amplifying equation (for ultra-short pulse widths), rather than a system appropriately the evanescent waves , and so on of coupled NLS equations (as in existing literature). The [1],[2]. Such metamaterials are experimentally realized by HNLS equation, which incorporates higher-order dispersive periodic arrays of thin conducting wires, exhibiting plasma and nonlinear terms, generalizes the one describing short pulse behavior, and split-ring resonators (SRR's), resembling paral­ propagation in nonlinear optical fibers. Analyzing the NLS lel plate capacitors, generating negative e and /i, respectively and HNLS equations, we find necessary conditions for the [3],[4]. Investigations of this class and other related types of formation of bright or dark solitons in the left-handed regime, metamaterials as well as various potential applications are and derive analytically approximate ultra-short solitons in non­ included in [5]-[10]. linear metamaterials. The research towards the investigation of So far, metamaterials have been mainly investigated in the above described phenomena was initiated in [14]. the linear regime, where e and /i are independent of the electric and magnetic field intensities. Nevertheless, nonlinear II. DESCRIPTION OF THE NONLINEAR metamaterials, which may be created by embedding an array Consider a lossless nonlinear metamaterial, characterized by of wires and SRR's into a nonlinear dielectric [11],[12], the effective permittivity and permeability [11], may prove useful in various applications. In particular, it has been demonstrated that the field intensity acts as a control 2 eH = e0(^(|E| )-^, (1) mechanism, altering the material properties from left- to right-handed and back. Hence, the study of metamaterials / FCJ2 \ nonlinear properties may prove useful in the implementation 2 fi(u) = /i0 1 2 2 7iTTi2\ ' < ) of tunable structures, with transmission controlled by the V ^2-^OTVL(IHI )/ field intensity, and in studying nonlinear effects in negative where UJP is the plasma , F is the filling factor, UJQNL refraction photonic crystals. Furthermore, it was shown in [13] is the nonlinear resonant SRR frequency [11], while E and H that left-handed weakly nonlinear (Kerr type) media support are the electric and magnetic field intensities, respectively. In propagation of vector solitons. the linear limit, €D —> 1 and UJQNL —> ^res (where ujres is

978-1-4577-1686-4/11/S26.00 ©2011 IEEE 75 the linear resonant SRR frequency), and left-handed behavior media (with e^ < 0 and JJLL < 0 - see Fig. 1), (ii) a may occurs in the frequency band ujres < UJ < mm{up,UM}, with be either positive or negative, while (iii) f3 is positive (for

UJM = ^res/Vl — F, provided that UJP > ujres. more details see the related discussion in [13]). Notice that, in Concerning the nonlinear properties of the metamaterial, we principle, €NL and J^NL may depend on both intensities |E|2 assume a weakly nonlinear (Kerr-type) behavior which can be and |H|2; such a case can also be studied via the analytical approximated by the decompositions [13]—[17]: approach described below.

2 2 e(^;|E| ) = eL(uj) + 6NL(\E\ ), (3) III. ELECTROMAGNETIC WAVE PROPAGATION We consider the propagation along the +z direction of a x- /i(^;|H|2) = /i (o;)+^L(|H|2), (4) L (y-) polarized electric (magnetic) field, namely, where the respective linear parts are given by E(z,t)=£E(z,t) , U(z,t) = yH(z,t). (9) CL(W) = eofl-^J, (5) Then, using the constitutive relations (in frequency domain) D=eE and B=/iH (D and B are the electric flux density and the magnetic induction), Faraday's and Ampere's laws respectively read (in the time domain): flL(u) = /i0 ( 1 2 2~ ' ^

dzE = -dt(n*H), dzH = -dt(e*E), (10) while the nonlinear parts of the permittivity and permeability depend linearly on the electric and magnetic field power where * denotes the convolution integral, i.e., f(t) * g(t) = respectively and are given by [13]—[17]: I-™ f(r)d(t ~ r)dr, for functions f(t) and g(t). Note that Eqs. (10) may be used in either the right-handed or e (\E\2) = e a|E|2, (7) NL 0 the left-handed regime of a metamaterial: once the dispersion 2 2 WVL(|H| ) = Mo/3|H| , (8) relation fco = ^o(^o) (connecting the carrier wavenumber fco and the carrier frequency UJO) and the evolution equations 2 where, a = ±E~ and (3 are the Kerr coefficients for for the fields E and H are found, then ko > 0 (fco < 0) the electric and magnetic fields, respectively, Ec being a corresponds to the right- (left-) handed regime. Alternatively, characteristic electric field value (for example of the order of for fixed fco > 0, one should shift the fields as [E,H]T —>> 200 V/cm for n-InSb [18]). [±E^H]T (either up or down sign combinations), thus The approximations (3)-(4) are physically justified consid­ inverting the orientation of the magnetic field and associated ering that the slits of the SRRs are filled with a nonlinear . Below, in our consideration we will assume dielectric [11],[18]. Generally, both cases of focusing and that the wavenumber fco = fco(^o) obtained from the linear defocusing dielectrics (corresponding, respectively, to a > 0 dispersion relation [see Eq. (22) below] will be fco < 0 for the and a < 0) are possible. The magnetic Kerr coefficient f3 left-handed regime. can be found via the dependence of /i on the magnetic field Next, we consider that the fields are expressed as intensity [11],[18]. T T Here, fixing the filling factor F = 0.4 and the plasma [E(z, t), H(z, t)] = [q(z, t),p(z, t)] exp[i(k0z - u0t)], (11) frequency UJP = 2TT x 10 GHz, we will perform our analysis in the frequency band 2TT X 1.45 GHz < uo < 2TT X 1.87 GHz, where q and p are the unknown electric and magnetic field inside which (i) the SRRs are negative-index left-handed envelopes, respectively.

IV. REDUCTIVE PERTURBATION METHOD Nonlinear evolution equations for the field envelopes can be found by the reductive perturbation method [19] as follows. First, we assume that the temporal spectral width of the nonlinear term with respect to the spectral width of the quasi- plane-wave dispersion relation is characterized by the formal small parameter e [20]-[22]. Then, we introduce the slow variables:

2 Z = e z, T = e(t-k'0z), (12) x where fc0 = v~ is the inverse of the group velocity (hereafter, primes will denote derivatives with respect to UJQ). Addition­ ally, we express q and p as asymptotic expansions in terms of Fig. 1. The linear parts of the relative magnetic permeability, HL/HO the parameter e, [solid (red) line], and the electric permittivity, CL/CO [dashed (blue) line] q(Z,T) =qo(Z,T)+s (Z,T) + s2q (Z,T) + • • • , (13) as functions of frequency, for F = 0.4 and UJP = 2TT X 10 GHz. In the band qi 2 cj = 27T x 1.45 GHz to UJM = ITX X 1.87 GHz both /zj, and ex, are res 2 negative and, thus, the medium is left-handed. p(Z,T) =Po(Z,T) + ePl(Z,T) + s p2(Z,T) + • • • , (14)

76 and assume that the Kerr coefficients a and j3, characterizing using Eq. (21), Eq. (16) suggests that the unknown vector xi the nonlinear parts of the dielectric permittivity and magnetic has the form, permeability, are of order 0(e2) (see, e.g., [13],[20] as well as [23],[24] for a corresponding analysis in ). xi = iR'dr^Z, T) + R^(Z, T), (24) Substituting Eqs. (13)—(14) into Eqs. (10), using Eqs. (3), where I/J(Z,T) is an unknown scalar field. Next, at order (4), and (12), and Taylor expanding the functions ei, and /x^,, G(e2), the compatibility condition for Eq. (17), combined with we arrive at the following equations at various orders of e: Eqs. (21) and (24), yields the following nonlinear Schrodinger 0 (NLS) equation, O(e ) Wx0 0, (15) l Wxi (16) 1 0(e ) -iW'drXo, 2 k0oT() ■7H 0, (25) 2 0(e ) Wx2 W"^x0 where fcg ^s me group-velocity dispersion (GVD) coefficient, 1 (17) as can be evaluated by differentiating k'0 in Eq. (23), and the nonlinear coefficient 7 is given by: 3 0{e ): Wx3 W"'<9f,x0 (26) ^'d^ + h'jd^-idzxi 6 2 It is important to note that once (j) is obtained from the (18) Axi + iBxo, NLS Eq. (25), the electric and magnetic field envelopes are respectively determined as qo =

= (j)-\-eijj. This way, required for Eqs. (15)—(18) to be solvable, known also as combining the NLS equations obtained at orders 0(e2) and Fredholm alternatives [19],[20], are LWx^ = 0, where L = 0(e3), we find that $ obeys the higher-order NLS (HNLS) [1,ZL] is a left eigenvector of of W, such that LW = 0, with equation: ZL — \/^L/CL being the linear wave-impedance. 1. The leading-order Eq. (15) provides the following results. 20z$--fco0T$ + 7l$l $ First, the solution XQ of Eq. (15) has the form: 2 xo = R0(Z,T), (21) -Kd^-^dTm ^) (27) where 0(Z, T) is an unknown scalar field (to be determined For e = 0, the HNLS Eq. (27) is reduced to the NLS Eq. 1 1 below) and R = [1, Z^ ] - is a right eigenvector of W, such (25), while for e 7^ 0 generalizes the higher-order NLS equa­ that WR = 0. Second, by using the compatibility condition tion describing ultra-short pulse propagation in optical fibers LWxo = 0 and Eq. (21), we obtain the equation LWR = 0, [20]-[22] (where /i = /JLQ, while dispersion and nonlinearity which is actually the linear dispersion relation, appear solely in the fiber dielectric properties). As in the case of the NLS Eq. (25), Eq. (27) provides the field $ which, k2 (22) uJ0eLfiL, in turn, determines the electric and magnetic fields at order 3 1 with all functions of frequency, e.g., CL and JUL are evaluated (D(e ), respectively, as qo + eq\ = & and po + ep\ = Z^ ^ at CJO- Note that Eq. (22) is also obtained by imposing the [see Eqs. (21) and (24)]. Finally, we stress that the NLS Eq. nontrivial solution condition detW = 0. Third, the electric (25), or the HNLS Eq. (27), can be used in the left- (right-) and magnetic field envelopes are proportional to each other, handed regime, taking ko, CL, and /JLL negative (positive) as namely, q0 = p0ZL. per the discussion above. Next, at 0(e1), the compatibility condition for Eq. (16) results in LW'R = 0, written equivalently as: V. SOLITONS SOLUTIONS OF THE NLS EQUATION

f f Now, we analyze Eqs. (25) and (27) in more detail in order 2k0k'0 = ujl(eLn L + e LfiL) + 2u0eLfiL. (23) to derive analytically exact soliton solutions propagating in the

This is actually the definition of the group velocity vg = under consideration nonlinear left-handed metamaterial. First, 1//CQ, as can also be found by differentiating the linear measuring length, time, and the field intensity \(j)\2 in units dispersion relation Eq. (22) with respect to UJ. Furthermore, of the dispersion length Lp = ^o/l^o l» initial pulse width to,

77 TABLE I [25], such solitons can be found upon seeking travelling-wave CONDITIONS FOR THE FORMATION OF BRIGHT OR DARK SOLITONS (BS solutions of Eq. (29) of the form, OR DS) FOR THE NLS EQ. (28). $(Z, T) = U{rj) exp[i(KZ - fiT)], (31) S = +l 8 = -1 C7 = +l a > 0 DS BS where U(rj) is the unknown envelope function (assumed to be real), r\ = T—AZ, and the real parameters A, K and ft denote, a = -l «f£ BS DS respectively, the inverse velocity, wavenumber and frequency of the travelling wave. Substituting Eq. (31) into Eq. (29), the C7 = +l a<0,\f\<§L DS BS real and imaginary parts of the resulting equation respectively read: K \fl2 -5 ff U x U U3 = 0, (32) and Lo/l'yl, respectively, we reduce the NLS Eq. (25) to the 35ift following dimensionless form: 2 SxU + (A - sfl - 36^)11 - 3aS2U U = 0, (33) idz4> ~ 2^T"5 ■ CT|0|S 0, (28) where overdots denote differentiations with respect to 77. where s = sign(fco) and & — sign(7)- The NLS Eq. (28) Notice that in the case of Si = S2 = 0, Eq. (33) is admits bright (dark) soliton solutions for sa = — 1 (sa = +1). automatically satisfied if A = sfl and the profile of "long" For our choice of parameters, numerical simulations indicate soliton pulses [governed by Eq. (28)] is determined by Eq. (see Fig. 2 of [14] for more details) that s = +1 (i.e., k'0' > 0) (32). On the other hand, for ultra-short solitons (corresponding for 2TT x 1.76 < u < 2TT X 1.87 GHz, while s = -1 (i.e., A# < to Si 7^ 0,^2 7^ 0), the system of Eqs. (32) and (33) is 0) for 27r x 1.45 < UJ < 2ir x 1.76 GHz inside the left-handed consistent if the following conditions hold: regime. As concerns the parameter a, it can take either the K \fl2 -Sifl3 A-sfl-3Sifl2 value a = +1 or a = —1, depending on the magnitudes and (34) signs of the Kerr coefficients a and f3. As mentioned above, 3Sifl Si here let us recall that we have assumed that (3 > 0, and hence we have a = +1 either for a focusing dielectric, with a > 0, 0S2 a(l + S fl) 2 (35) or for a defocusing dielectric, a < 0, with \a//3\ < Z^jZ\ ' f +3SifL v, (ZQ = y/W^o is the vacuum wave-impedance). where K and v are nonzero constants. In such a case, Eqs. (32) Thus, for a = +1, bright (dark) solitons occur in the and (33) are equivalent to the following equation of motion of anomalous (normal) dispersion regimes, namely, for k ' < 0 0 the unforced and undamped Duffing oscillator, (fco > 0)' respectively. On the other hand, a = — 1 for s a defocusing dielectric (a < 0), with \a//3\ > ZQ/Z^ U + KU + vU = 0. (36) and, bright (dark) solitons occur in the normal (anomalous) dispersion regimes. The above results are summarized in Table For KV < 0, Eq. (36) possesses two exponentially localized I. solutions (as special cases of its general elliptic function Importantly, note that the "flexibility" arising from the extra solutions), corresponding to the separatrices in the (U,U) "degree of freedom" provided by the presence of dispersion phase-plane of the dynamical system at hand [25]. These and nonlinearity properties in the magnetic response of the solutions have the form of a hyperbolic secant (tangent) for left-handed metamaterial (missing in fiber optics), allows for K < 0 and v > 0 (K, > 0 and v < 0), thus corresponding to the formation of bright (dark) solitons in the anomalous the bright, UBs (dark, J7DS) solitons of Eq. (29): (normal) dispersion regimes for defocusing dielectrics (see 1/2 UBS(v) = (2H/^ sech(^77), (37) third line of Table I). 1/2 VI. ULTRA-SHORT SOLITONS SOLUTIONS OF THE HNLS Uvs(v) (2«/|i/|) tanh(, (38) EQUATION Importantly, these are ultra-short solitons of the HNLS Eq. Next, we consider the HNLS Eq. (27) which, by using the (29), valid even for e = 0(1): since both coefficients Si, S2 of same dimensionless units as before, is expressed as, Eq. (29) scale as ^(CJO^O)-1? it is clear that for CJO^O = 0(1), or for soliton widths to ~ CJ^-1, the higher-order terms can id $ - |<9|$ + a|$|2$ = iS^Q - ia£ <9 (|$|2$), (29) z 2 T safely be neglected and soliton propagation is governed by where the coefficients Si and S2 are given by: Eq. (28). On the other hand, if ujoto = O(e), the higher-order terms become important and solitons governed by the HNLS 1 1 Si (30) Eq. (29) are ultra-short, of a width to ~ SUQ . We stress '6£0|A#|' ^0*0' that these solitons are approximate solutions of Maxwell's Equation (29) can be used to predict ultra-short solitons in equations, satisfying Faraday's and Ampere's laws in Eqs. (10) nonlinear left-handed metamaterials. More precisely, following up to order G(e3).

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