Nonlinear Wave Propagation in Negative Index ...

Nonlinear Wave Propagation in Negative Index Metamaterials Nikolaos L. Tsitsas 1 and Dimitri J. Frantzeskakis 2 1

School of Applied Mathematical and Physical Sciences, National Technical University of Athens, Zografos, Athens 15773, Greece 2 Department of Physics, University of Athens, Panepistimiopolis, Zografos, Athens 15784, Greece [email protected] Abstract— Wave propagation in nonlinear negative index metamaterials is investigated by directly implementing the reductive perturbation method to Faraday's and Ampere's laws. In this way, we derive a second-order and a third-order nonlinear Schrodinger equation, describing solitons of moderate and ultrashort pulse widths, respectively. Wefindnecessary conditions and derive exact bright and dark soliton solutions of these equations for the electric and magnetic field envelopes. Directions of future work towards the modelling of wave propagation in more complicated types of nonlinear negative index metamaterials (e.g., chiral metamaterials) are pointed out. I. INTRODUCTION

Metamaterials possessing negative refractive index, due to simultaneous negative permittivity e and permeability /i, have become a subject of intense research activity, since they are characterized by remarkable electromagnetic properties, including: negative refraction for interface scattering (reversal of Snell's law), backward wave propagation, reversal of the Doppler shift and Cherenkov effect, collecting lens behavior forming 3-D images, perfect lens performance by amplifying appropriately the evanescent waves amplitudes, and so on [1],[2]. Such metamaterials are experimentally realized by periodic arrays of thin conducting wires, exhibiting plasma behavior, and split-ring resonators (SRR's), resembling paral­ lel plate capacitors, generating negative e and /i, respectively [3],[4]. Investigations of this class and other related types of metamaterials as well as various potential applications are included in [5]-[10]. So far, metamaterials have been mainly investigated in the linear regime, where e and /i are independent of the electric and magnetic field intensities. Nevertheless, nonlinear metamaterials, which may be created by embedding an array of wires and SRR's into a nonlinear dielectric [11],[12], may prove useful in various applications. In particular, it has been demonstrated that the field intensity acts as a control mechanism, altering the material properties from left- to right-handed and back. Hence, the study of metamaterials nonlinear properties may prove useful in the implementation of tunable structures, with transmission controlled by the field intensity, and in studying nonlinear effects in negative refraction photonic crystals. Furthermore, it was shown in [13] that left-handed weakly nonlinear (Kerr type) media support propagation of vector solitons.

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In this work, we start from Maxwell's equations, and use the reductive perturbation method to derive a second-order and a third-order nonlinear Schrodinger (NLS) equation, describing ultra-short solitons in nonlinear left-handed metamaterials. Then, we find necessary conditions and derive exact bright and dark soliton solutions of these equations for the electric and magnetic field envelopes. More precisely, we present a systematic derivation of NLS and higher-order NLS (HNLS) equations for the electromag­ netic field envelopes, as well as ultra-short solitons for lefthanded metamaterials. In particular, we use the reductive per­ turbation method to derive from Faraday's and Ampere's laws a hierarchy of equations. Using such an approach, i.e., directly analyzing Maxwell's equations, rather than coupled wave equations for the electromagnetic field envelopes, we show that the electric field envelope is proportional to the magnetic field one (their ratio being the linear wave-impedance). Thus, for each of the electromagnetic wave components, we derive a single NLS (for moderate pulse widths) or a single HNLS equation (for ultra-short pulse widths), rather than a system of coupled NLS equations (as in existing literature). The HNLS equation, which incorporates higher-order dispersive and nonlinear terms, generalizes the one describing short pulse propagation in nonlinear optical fibers. Analyzing the NLS and HNLS equations, we find necessary conditions for the formation of bright or dark solitons in the left-handed regime, and derive analytically approximate ultra-short solitons in non­ linear metamaterials. The research towards the investigation of the above described phenomena was initiated in [14]. II. DESCRIPTION OF THE NONLINEAR METAMATERIAL

Consider a lossless nonlinear metamaterial, characterized by the effective permittivity and permeability [11],

eH

=

e0(^(|E|2)-^,

fi(u)

=

/i 0 1

FCJ 2

/

V

(1)

2 2

2

\

7iTTi2\ '

1 and UJQNL —> ^res (where ujres is

75

the linear resonant SRR frequency), and left-handed behavior occurs in the frequency band ujres < UJ < mm{up,UM}, with UJM = ^ r e s / V l — F, provided that UJP > ujres. Concerning the nonlinear properties of the metamaterial, we assume a weakly nonlinear (Kerr-type) behavior which can be approximated by the decompositions [13]—[17]: e(^;|E|2)

=

eL(uj) + 6NL(\E\2),

(3)

/i(^;|H|2)

=

/iL(o;)+^L(|H|2),

(4)

media (with e^ < 0 and JJLL < 0 - see Fig. 1), (ii) a may

be either positive or negative, while (iii) f3 is positive (for more details see the related discussion in [13]). Notice that, in principle, €NL and J^NL may depend on both intensities | E | 2 and | H | 2 ; such a case can also be studied via the analytical approach described below. III.

We consider the propagation along the + z direction of a x(y-) polarized electric (magnetic) field, namely,

where the respective linear parts are given by

E(z,t)=£E(z,t)

CL(W) = e o f l - ^ J , flL(u)

=

/i0 ( 1

2

'

^

dzE = -dt(n*H),

while the nonlinear parts of the permittivity and permeability depend linearly on the electric and magnetic field power respectively and are given by [13]—[17]: eNL(\E\2) WVL(|H|2)

=

e0a|E|2,

=

Mo/3|H|

2

U(z,t)

= yH(z,t).

(9)

dzH = -dt(e*E),

(10)

where * denotes the convolution integral, i.e., f(t) * g(t) = 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 the left-handed regime of a metamaterial: once the dispersion relation fco = ^o(^o) (connecting the carrier wavenumber fco and the carrier frequency UJO) and the evolution equations for the fields E and H are found, then ko > 0 (fco < 0) corresponds to the right- (left-) handed regime. Alternatively, for fixed fco > 0, one should shift the fields as [E,H]T —>> [±E^H]T (either up or down sign combinations), thus inverting the orientation of the magnetic field and associated Poynting vector. Below, in our consideration we will assume that the wavenumber fco = fco(^o) obtained from the linear dispersion relation [see Eq. (22) below] will befco< 0 for the left-handed regime. Next, we consider that the fields are expressed as

(7) ,

,

Then, using the constitutive relations (in frequency domain) D = e E and B = / i H ( D and B are the electric flux density and the magnetic induction), Faraday's and Ampere's laws respectively read (in the time domain):

(5) 2~

ELECTROMAGNETIC WAVE PROPAGATION

(8)

2

where, a = ±E~ and (3 are the Kerr coefficients for the electric and magnetic fields, respectively, Ec being a characteristic electric field value (for example of the order of 200 V/cm for n-InSb [18]). The approximations (3)-(4) are physically justified consid­ ering that the slits of the SRRs are filled with a nonlinear dielectric [11],[18]. Generally, both cases of focusing and defocusing dielectrics (corresponding, respectively, to a > 0 and a < 0) are possible. The magnetic Kerr coefficient f3 can be found via the dependence of /i on the magnetic field intensity [11],[18]. [E(z, t), H(z, t)]T = [q(z, t),p(z, t)]T exp[i(k0z - u0t)], Here, fixing the filling factor F = 0.4 and the plasma (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. R E D U C T I V E P E R T U R B A T I O N M E T H O D

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 quasiplane-wave dispersion relation is characterized by the formal small parameter e [20]-[22]. Then, we introduce the slow variables: Z = e2z,

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 the parameter e,

Fig. 1. The linear parts of the relative magnetic permeability, HL/HO [solid (red) line], and the electric permittivity, CL/CO [dashed (blue) line] as functions of frequency, for F = 0.4 and UJP = 2TT X 10 GHz. In the band cj r e s = 27T x 1.45 GHz to UJM = ITX X 1.87 GHz both /zj, and ex, are negative and, thus, the medium is left-handed.

76

q(Z,T)

=qo(Z,T)+sqi(Z,T)

+ s2q2(Z,T)

+ • • • , (13)

p(Z,T)

=Po(Z,T)

+ s2p2(Z,T)

+ • • • , (14)

+ ePl(Z,T)

and assume that the Kerr coefficients a and j3, characterizing the nonlinear parts of the dielectric permittivity and magnetic permeability, are of order 0(e2) (see, e.g., [13],[20] as well as [23],[24] for a corresponding analysis in optics). Substituting Eqs. (13)—(14) into Eqs. (10), using Eqs. (3), (4), and (12), and Taylor expanding the functions ei, and /x^,, we arrive at the following equations at various orders of e:

O(e0) 0(el) 2

0(e )

Wx0

(15)

0,

Wxi

W"^x0

Wx2

(17)

+

6

W = B x ,o

(24)

where I/J(Z,T) is an unknown scalar field. Next, at order G(e2), the compatibility condition for Eq. (17), combined with Eqs. (21) and (24), yields the following nonlinear Schrodinger (NLS) equation, 1

k0oT()

■7H2

0,

(25)

where fcg ^s m e group-velocity dispersion (GVD) coefficient, as can be evaluated by differentiating k'0 in Eq. (23), and the nonlinear coefficient 7 is given by:

a

(26)

h'jd^-idzxi 2 (18)

A x i + iBxo, [QhPi]

T) + R ^ ( Z , T ) ,

W"' ~ 2^T"5

0,

■ CT|0| S

ULTRA-SHORT SOLITONS SOLUTIONS OF THE EQUATION

HNLS

- ia£ 2 0 (K, > 0 and v < 0), thus corresponding to the bright, UBs (dark, J7DS) solitons of Eq. (29): UBS(v)

=

(2H/^1/2sech(^77), (2«/|i/|) 1 / 2 tanh(,

(37)

(38)

Importantly, these are ultra-short solitons of the HNLS Eq. (29), valid even for e = 0(1): since both coefficients Si, S2 of 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 safely be neglected and soliton propagation is governed by Eq. (28). On the other hand, if ujoto = O(e), the higher-order terms become important and solitons governed by the HNLS Eq. (29) are ultra-short, of a width to ~ SUQ1 . We stress that these solitons are approximate solutions of Maxwell's equations, satisfying Faraday's and Ampere's laws in Eqs. (10) up to order G(e3).

where the coefficients Si and S2 are given by: Si

-5xff

35ift

Uvs(v)

Next, we consider the HNLS Eq. (27) which, by using the same dimensionless units as before, is expressed as, idz$ - | < 9 | $ + a | $ | 2 $ = iS^Q

\fl2

K

SxU + (A - sfl - 36^)11

(28)

where s = sign(fco) a n d & — s ig n (7)- The NLS Eq. (28) admits bright (dark) soliton solutions for sa = — 1 (sa = +1). For our choice of parameters, numerical simulations indicate (see Fig. 2 of [14] for more details) that s = +1 (i.e., k'0' > 0) for 2TT x 1.76 < u < 2TT X 1.87 GHz, while s = - 1 (i.e., A# < 0) for 27r x 1.45 < UJ < 2ir x 1.76 GHz inside the left-handed regime. As concerns the parameter a, it can take either the value a = +1 or a = —1, depending on the magnitudes and signs of the Kerr coefficients a and f3. As mentioned above, 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, or for a defocusing dielectric, a < 0, with \a//3\ < Z^jZ\ (ZQ = y / W ^ o is the vacuum wave-impedance). Thus, for a = + 1 , bright (dark) solitons occur in the anomalous (normal) dispersion regimes, namely, for k0' < 0 (fco > 0)' respectively. On the other hand, a = — 1 for a defocusing dielectric (a < 0), with \a//3\ > ZQ/Z^ and, bright (dark) solitons occur in the normal (anomalous) dispersion regimes. The above results are summarized in Table I. Importantly, note that the "flexibility" arising from the extra "degree of freedom" provided by the presence of dispersion and nonlinearity properties in the magnetic response of the left-handed metamaterial (missing in fiber optics), allows for the formation of bright (dark) solitons in the anomalous (normal) dispersion regimes for defocusing dielectrics (see third line of Table I). VI.

(31)

(30)

Equation (29) can be used to predict ultra-short solitons in nonlinear left-handed metamaterials. More precisely, following

78

Finally, concerning the condition for bright or dark soliton formation, namely KV < 0, we note that K depends on the free parameters K and ft (and, thus, can be tuned on demand), while the parameter v has the opposite sign from a (since 82 > 0, while sign(5i) = sign(fco') — +!)• This means that bright solitons are formed for K, < 0 and a = — 1 (i.e., a < 0 with \a//3\ > ZQ/Z^), while dark ones are formed for K > 0 and a = +1 (i.e., a > 0, or a < 0 with \a//3\ < Z$/Z%). VII.

CONCLUSIONS AND FUTURE WORK

We used the reductive perturbation method to derive from Maxwell's equations a nonlinear Schrodinger (NLS) and a higher-order NLS (HNLS) equation describing pulse propa­ gation in nonlinear metamaterials. We studied in detail the pertinent dispersive and nonlinear effects, and found necessary conditions for the formation of either bright or dark ultra-short solitons, as well as approximate analytical expressions for these solutions, in the left-handed regime of the metamaterial. Interesting subjects for future research may include a sys­ tematic study of the stability and dynamics of the ultrashort solitons, both in the framework of the HNLS equation and, perhaps more importantly, in the context of Maxwell's equations. Moreover, it is worth to carry out the corresponding in­ vestigations, concerning solitons formation and propagation, in other more complicated types of nonlinear metamaterials, exhibiting negative refractive index in a certain frequency band. For example a representative material of this class may be an isotropic chiral metamaterial for which one of the two refractive indices can have a negative real part [26], [27]. In the case of a nonlinear chiral metamaterial, application of the reductive perturbation method would lead to a system of two coupled NLS equations for the left-handed and righthanded Beltrami components of the electromagnetic field. For a sufficiently large chirality parameter there exists a certain spectral regime where the refractive index for the left/righthanded Beltrami component is real and negative but that for the right/left-handed Beltrami component is real and positive [28]. To this direction, it would be interesting to approximate, inside the above regime, the coupled system of the NLS equations by the Manakov system [29] which is known to be completely integrable, and hence predict various classes of exact vector soliton solutions (bright-bright, dark-dark, as well as dark-bright solitons) that can be supported in the nonlinear chiral metamaterial. Relevant investigations are in process and will be reported in a future publication. REFERENCES [1] V. G. Veselago, "Electrodynamics of substances with simultaneously negative values of sigma and mu," Sov. Phys. Usp., vol. 10, p. 509, 1968. [2] J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett., vol. 85, p. 3966, 2000. [3] D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative perme­ ability and permittivity," Phys. Rev. Lett., vol. 84, p. 4184, 2000. [4] A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science, vol. 292, p. 77, 2001.

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