## Abstract

We theoretically prove that electromagnetic beams propagating through a nonlinear cubic metamaterial can exhibit a power flow whose direction reverses its sign along the transverse profile. This effect is peculiar of the hitherto unexplored extreme nonlinear regime where the nonlinear response is comparable or even greater than the linear contribution, a condition achievable even at relatively small intensities. We propose a possible metamaterial structure able to support the extreme conditions where the polarization cubic nonlinear contribution does not act as a mere perturbation of the linear part.

© 2010 Optical Society of America

## 1. Introduction

The ability of manufacturing metamaterials with prescribed and anomalous values of permittivity *ε* and permeability *μ* has triggered an intense research effort aimed at investigating novel regimes of linear electromagnetic propagation and suitable configurations have been devised for observing remarkable effects such as, for example, superlensing [1], optical cloaking [2], guiding of nanometric optical beams [3] and photonic circuits [4]. In the nonlinear realm, the nonlinear properties of left-handed metamaterials have been investigated [5] together with various soliton manifestations [6]. Propagation in metamaterials exhibiting cubic nonlinear response has also been considered [7] and, for ultra-short pulse nonlinear dynamics, it has been suggested that metamaterial linear property tailoring allows the observation of different nonlinear regimes [8].

In this Letter we show that a metamaterial with a very small linear dielectric constant and exhibiting a nonlinear cubic response is able to support nonlinear guided waves whose Poynting vector has the very peculiar property of being parallel and anti-parallel to the propagation direction in different transverse portion of the field. This novel phenomenology is a consequence of the fact that, since the metamaterial linear dielectric permittivity can be arbitrary small, the nonlinear contribution to the dielectric response can easily (i.e. at low intensities) be made comparable or greater than the linear part so that, the sign of the overall dielectric response can be different for different intensities. In the presence of an electromagnetic beam this implies that conditions can be found so that the effective dielectric response has different signs on the propagation axis and at its lateral sides. Therefore the transverse reversing of the power flow is understood since, for a monochromatic Transverse Magnetic (TM) field mainly propagating along a given direction, the Poynting vector globally lies along the same mean propagation direction and its sign coincides with that of the total effective dielectric constant. In order to discuss this effect on a feasible situation, we consider TM electromagnetic propagation in a defocusing nonlinear cubic metamaterial and we analytically obtain a class of nonlinear guided waves exhibiting the aforementioned transverse power flow reversing. It is remarkable that the power flow reversing effect can be observed even at very low intensities since it is a consequence of the interplay between the linear and nonlinear contributions to the dielectric response, the former being very small in the considered metamaterials and the latter being proportional to the intensity. The question naturally arises as to whether a medium exists or can be conceived where the range of electromagnetic intensities, for which its nonlinear response is purely cubic, is so large to produce a huge nonlinear response. At first one may reject this possibility since the cubic nonlinear response generally arises from a perturbative description of radiation-matter interaction so that the nonlinear polarization necessarily is a small correction to the linear part. However, exploiting the availability of metamaterials with somehow prescribed values of the dielectric permittivity, we propose that in a suitable sub-wavelength layered structure, consisting of alternating slabs of a metamaterial with negative dielectric constant and an optically active nonlinear cubic medium, the effective electromagnetic response is purely cubic in an intensity range where the nonlinear cubic term can exceed the linear contribution.

## 2. Nonlinear guided waves

Consider a monochromatic electromagnetic field (whose time variation is assumed to be *e*
^{-iωt}, where *ω* is the angular frequency) propagating through a nonlinear metamaterial characterized by the constitutive relations (holding for the field complex amplitudes)

$$\mathbf{B}={\mu}_{0}\mu \mathbf{H},$$

where *ε* > 0 and *μ* > 0 are the linear permittivity and permeability, respectively, whereas *χ* > 0 and 0 < *γ* < 1 are the parameters characterizing the cubic defocusing nonlinear response. We focus our attention on transverse magnetic (TM) nonlinear guided waves propagating along the *z*- axis of the form

$$\mathbf{H}\left(x,z\right)={e}^{\mathrm{i\beta \zeta}}\sqrt{\frac{{\epsilon}_{0}{\epsilon}^{2}}{{\mu}_{0}\mathrm{\mu \chi}}}\phantom{\rule{.3em}{0ex}}\left[\beta {u}_{x}\left(\xi \right)-\frac{d{u}_{z}\left(\xi \right)}{\mathrm{d\xi}}\right]\phantom{\rule{.3em}{0ex}}{\hat{\mathbf{e}}}_{y}$$

where $\xi =\sqrt{\mathrm{\epsilon \mu}}(\omega /c)x,\phantom{\rule{.2em}{0ex}}\zeta =\sqrt{\mathrm{\epsilon \mu}}(\omega /c)z$ (*c* is the speed of light in vacuum) are dimensionless spatial coordinates, *β* is a real dimensionless propagation constant and *u _{x}* and

*u*are dimensionless electric field components. Substituting the fields of Eqs. (2) into Maxwell equations ∇ ×

_{z}**E**=

*iω*

**B**and ∇ ×

**H**= -

*iω*

**D**and using the constitutive relations of Eqs. (1) we get

$$\frac{{d}^{2}{u}_{z}}{d{\xi}^{2}}-\beta \frac{d{u}_{x}}{\mathrm{d\xi}}=\left[-1+\left(1-\gamma \right){u}_{x}^{2}+\left(1+\gamma \right){u}_{z}^{2}\right]{u}_{z}$$

which is a system of ordinary differential equations fully characterizing the transverse profile of the considered nonlinear guided waves. Without loss of generality we consider solutions of Eqs. (3) with definite parity where *u _{x}* and

*u*are spatially even (

_{z}*u*(

_{x}*ξ*) =

*u*(-

_{x}*ξ*)) and odd (

*u*(

_{z}*ξ*) = -

*u*(-

_{z}*ξ*)), respectively and, as a consequence, we adopt the boundary conditions

*u*(0) =

_{x}*u*

_{x0},

*u*(0) = 0 and

_{z}*u*(+∞) =

_{x}*u*

_{x∞},

*u*(+∞) =

_{z}*u*

_{z∞}. Since

*u*(

_{x}*ξ*) and

*u*(

_{z}*ξ*) have to asymptotically approach two constant values, their first and second derivative vanish for

*ξ*→ +∞ so that, exploiting the boundary conditions, we require the right hand sides of Eqs. (3) to vanish at

*u*=

_{x}*u*

_{x∞}and

*u*=

_{z}*u*

_{z∞}. Therefore we obtain $\beta =\sqrt{2\gamma \left(1-2{u}_{x\infty}^{2}\right)/\left(1+\gamma \right)}$ and ${u}_{z\infty}=\sqrt{\left[1-\left(1-\gamma \right){u}_{x\infty}^{2}\right]/\left(1+\gamma \right)}$ from which we note that

*u*

^{2}

_{x∞}< 1/2 is a necessary condition for the existence of the considered nonlinear waves. In order to prove their existence, we exploit the fact that the system of Eqs. (3) is integrable [9] since it admits the first integral

$$\phantom{\rule{3.7em}{0ex}}-\frac{1}{{\beta}^{2}}{\left[\left({\beta}^{2}-1\right)+\left(1+\gamma \right){u}_{x}^{2}+\left(1-\gamma \right){u}_{z}^{2}\right]}^{2}{u}_{x}^{2}$$

or, in other words, the relation $\frac{d}{\mathrm{d\xi}}F({u}_{x}\left(\xi \right),{u}_{z}\left(\xi \right))=0$ holds for any solution *u _{x}*(

*ξ*),

*u*(

_{z}*ξ*) of Eqs. (3). Evidently, after substituting the obtained

*β*into Eq. (4),

*F*has a stationary point at (

*u*

_{x∞},

*u*

_{z∞}) and the guided waves are represented by curves of constat

*F*in the plane (

*u*,

_{x}*u*) joining (

_{z}*u*

_{x0}, 0) to the stationary point. Therefore, requiring that

*F*has a saddle point at (

*u*

_{x∞},

*u*

_{z∞}) and exploiting the above necessary condition, we conclude that the considered nonlinear waves exist in the range

which is always not empty since *γ* > 0. In addition the relation *F*(*u*
_{x0},0) = *F*(*u*
_{x∞},*u*
_{z∞}) yields the possible values of *u _{x}*(0) =

*u*

_{x0}corresponding to the asymptotical value

*u*(+∞) =

_{x}*u*

_{x∞}. In Fig. 1(a) and 1(b) we plot the profiles of

*u*and

_{x}*u*corresponding to different values of

_{z}*u*

_{x∞}, spanning the range of Eq. (5), for

*γ*= 0.5 obtained by numerically integrating Eqs. (3) with the above boundary conditions. The power flow carried by these waves is described by the time-average Poynting vector $\mathbf{S}=\frac{1}{2}\mathrm{Re}\left(\mathbf{E}\times {\mathbf{H}}^{*}\right)$ which, exploiting Eqs.(2) and the first of Eq. (3), becomes

i.e., for the considered waves, is purely along the *z*- axis. In Fig. 1(c) we plot the profiles of *S _{z}* evaluated for the fields reported in Fig. 1(a) and 1(b), and we note that the sign of

*S*is not constant along the transverse profiles, a region where

_{z}*S*< 0 (red portion of the curves) existing around

_{z}*ξ*= 0. This reversing of the power flow along the transverse profile of the nonlinear guided waves is particularly evident from Fig. 1(d) where we draw the vector field

**S**on the plane (

*ξ*,

*ζ*) for one of the fields of Fig. 1(a) and 1(b). In order to physically grasp and discuss this unusual effect we recast Eq. (6) in the form

where use of Eqs. (2) and (1) has been made, from which it is evident that the transverse power flow reversing is a consequence of the sign flipping of *D _{x}* along the wave profile while

*E*does not change its sign. This implies that, regardless the absolute sign of the fields, the overall effective dielectric response undergoes a sign reversing due to the fact that, in the first of Eq. (1), the nonlinear cubic term can be both smaller and greater than the linear part, depending on the local field strengths. It is worth noting that, although we have discussed this effect using the considered nonlinear guided waves admitting analytical treatment, the phenomenon is more general and holds for the wider class of TM fields of the form

_{x}**E**(

*x*,

*z*) =

*e*[

^{ikz}*E*(

_{x}*x*,

*z*)

**e**̂

_{x}+

*iE*(

_{z}*x*,

*z*)

**e**̂

_{z}] since, if ∣∂

_{z}

*E*∣ ≪

_{x}*k*∣

*E*∣ and ∣∂

_{x}_{z}

*E*∣ ≪

_{z}*k*∣

*E*∣ (i.e. the field manly propagates along the

_{z}*z*- axis) it is simple to obtain from Maxwell equations that ${S}_{z}=\frac{\omega}{2k}\mathit{Re}\left[{D}_{x}^{*}\left(x,z\right){E}_{x}\left(x,z\right)\right]$ and the transverse power flow reversing can take place through the just discussed mechanism. We conclude that the predicted power flow reversing is a signature of the extreme nonlinear regime where the cubic nonlinear contribution to the medium polarizability is not a mere perturbation of the linear part. In this sense the medium behaves as a metamaterial whose character (positive or negative dielectric constant) locally depends on the field intensity. The discussed power flow reversing should be compared with the inhomogeneous power flow distribution occurring in linear photonic crystals since there it is associated to the fact that the variation of the index of refraction is comparable to average index of refraction. It is worth stressing that the discussed power flow reversing is very different from the effect that, in left handed metamaterials, the Poynting vector is antiparallel to the carrier wave vector which is a consequence of the fact that, in such media,

*ε*< 0 and

*μ*< 0 (with

*n*< 0). On the other hand, in our case,

*μ*> 0 and the sign of the power flow is not uniform being controlled through the field intensity. Note that such an extreme condition can be achieved when the field intensity ∣

*E*∣

^{2}is comparable or greater than

*ε*/

*χ*, so that, in standard materials where

*ε*is generally of the order of unity and

*χ*is very small (of the order of 10

^{-20}m

^{2}/V

^{2}in semiconductors [10]), the required intensity is so large to rule out the whole discussed phenomenology. However, if a metamaterial is employed where

*ε*can be chosen to be much smaller than unity, the intensity threshold can be reduced to the point of making the extreme nonlinear regime accessible even for intensities much smaller than those employed in standard nonlinear optics experiments.

## 3. Nonlinear layered medium supporting the extreme nonlinear regime

Even though the use of a metamaterial (*ε* ≪ 1) makes feasible intensities able to trigger the above linear-nonlinear competition, the main issue remains of finding a medium whose dielectric response is, in the considered intensity range, purely cubic. In fact, the first of Eqs. (1) is a power series expansion of the constitutive relation *D* = *D*(*E*) in the field strength *E* and therefore, if the third order is comparable with the first one, one generally has to consider higher order terms. In order to show that the discussed extreme nonlinear regime can effectively be achieved, consider the metamaterial structure reported in Fig. 2 consisting of alternate linear metamaterial and nonlinear medium layers, along the *y*-axis, of thickness *d*
_{1} and *d*
_{2} respectively. The metamaterial is a negative dielectric (ND) whose constitutive relation is **D** = *ε*
_{0}
*ε*
_{1}
**E** (where *Re*(*ε*
_{1}) < 0) whereas the nonlinear medium (NL) is characterized by the constitutive relation **D** = *ε*
_{0}
*ε*
_{2}
**E** - *ε*
_{0}
*χ*
_{2}[(**E**·**E**
^{*})**E** + *γ*(**E**·**E**)**E**
^{*}], i.e. it is an isotropic defocusing (*R _{e}*(

*ε*

_{2}) > 0,

*χ*

_{2}> 0) Kerr medium. The ND medium is generally a metal so that, in order to compensate losses, we suppose that the NL medium is also an active medium and that

*Im*(

*ε*

_{2}) can be tuned by adjusting the pumping efficiency (see example below) [11]. The media relative permeability are [

*μ*

_{1}and

*μ*

_{2}, respectively. If the spatial period

*d*=

*d*

_{1}+

*d*

_{2}is much smaller than the field vacuum wavelength 2

*πc*/

*ω*, the TM field propagating through the structure experiences the effective response described by Eqs. (1) and characterized by the spatially averaged parameters

$$\chi \phantom{\rule{.2em}{0ex}}=\phantom{\rule{.2em}{0ex}}\left(1-f\right){\chi}_{2},$$

$${\mu}^{-1}\phantom{\rule{.2em}{0ex}}=\phantom{\rule{.2em}{0ex}}f{\mu}_{1}^{-1}+\left(1-f\right){\mu}_{2}^{-1}$$

where *f* = *d*
_{1}/(*d*
_{1}+*d*
_{2}) is the fraction of negative dielectric. From these relations, it is evident that suitable values of *ε*
_{1}, *f* and *Im*(*ε*
_{2}) can be chosen so that 0 < *Re*(*ε*) ≪ 1 and ∣*Im*(*ε*)∣ ≪ *Re*(*ε*) i.e. the overall medium effective response coincides with the one considered in present Letter. Most importantly, the medium is able to support the extreme nonlinear regime since if the field is such that ∣*E*∣^{2} ~ *ε*
_{2}/*χ*, at the same time one has that ∣*E*∣^{2} ≪ *ε*
_{2}/*χ*
_{2}. Therefore the nonlinear medium layers (NL) are in the presence of a field for which their response is purely cubic and, as a consequence, the overall averaged structure response is purely cubic as well.

As a specific example, consider a TM field of wavelength *λ* = 0.810 *μm* propagating through a layered metamaterial structure for which *ε*
_{1} = -28.79800 + 1.55000*i*, *μ*
_{1} = 1 and *ε*
_{2} = 10.90000 - 0.56750*i*, *μ*
_{2} = 1, *χ*
_{2} = 6.56 × 10^{-18}
*m*
^{2}/*V*
^{2}, *γ* = 0.5. For the considered wavelength *λ*, the chosen *ε*
_{1} coincides with the silver permittivity [12] whereas *Re*(*ε*
_{2}) and *χ*
_{2} are the linear and nonlinear parameters characterizing the AlGaAs [13]. Here we are exploiting the fact that AlGaAs optically amplifies the radiation at the chosen wavelength if the sample is pumped by ultra-violet light and, consequently, the above value of *Im*(*ε*
_{2}) can be attained simply by adjusting the pump laser intensity [7, 14]. Choosing *f* = 0.2754, from Eqs. (8) we obtain the effective medium parameters *ε* = 0.00235 + 0.00003*i*, *μ* = 1 and *χ* = 4.76 × 10^{-18}m^{2}/*V*
^{2}. The absorption coefficient of the considered effective medium is *α* = (4*π*/*λ*)*Im*(∞*ε*) ≃ 4×10^{-3}
*μm*
^{-1} so that the above power flow reversing effect can be observed for propagation distances up to the decay length 1/*α* ≃ 208 *μm* (note that this decay length can be made larger by improving the balance between losses and gain). Consider now the nonlinear guided wave whose power flow is reported in Fig. (2)b which is characterized by *u*
_{x∞} = 0.65 and a transverse dimensionless width ∆*ξ* ≃ 2 (∆*ξ* also coincides with the width of the transverse portion of the field where the Poynting vector is antiparallel to the propagation direction). The physical width of the considered wave is $\Delta x=(\lambda /2\pi )\left(\Delta \xi /\sqrt{\mathrm{Re}\left(\epsilon \right)}\right)\simeq 5.3\mu m$ whereas the maximum of its normalized Poynting vector is *S _{z}*/

*S*

_{0}≃ 0.1 (see Fig. 2) so that the wave is characterized by the intensity

*S*= 0.1

_{z}*S*

_{0}≃ 0.3

*MW*/

*cm*

^{2}. It is worth stressing that the considered micron-sized confined wave is observable with an intensity (~

*MW*/

*cm*

^{2}) much smaller than that (~

*GW*/

*cm*

^{2}) required for exciting a spatial soliton (of the same width and at the same wavelength) propagating through a AlGaAs sample [15].

## 4. Conclusions

In conclusion, we have shown that nonlinear metamaterial with very small dielectric permittivity can support the propagation of electromagnetic beams exhibiting transverse power flow reversing, i.e. the Poynting vector changes sign along their transverse profile. Such an unusual phenomenon is one of the manifestations of the underlying extreme nonlinear regime where the nonlinear contribution to the polarizability can even exceed the linear contribution, a situation never occurring in general nonlinear optical setups. Therefore we conclude that the designing of complex metamaterials hosting the discussed extreme nonlinear regime can play a fundamental role for conceiving sub-wavelength nonlinear devices since very high confinements are achievable with low intensities.

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