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Keywords:

  • metamaterials;
  • nonlinearity;
  • solitons

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Nonlinear Resonant Response
  5. 3. FDTD Simulations of Nonlinear Transmission
  6. 4. Electromagnetic Spatial Solitons
  7. 5. Conclusions
  8. Acknowledgments
  9. References

[1] We analyze nonlinear properties of microstructured materials with the negative refractive index, the so-called left-handed metamaterials. We demonstrate that the hysteresis-type dependence of the magnetic permeability on the field intensity allows changing the material properties from left to right handed and back. Using the finite difference time domain simulations, we study the wave reflection from a slab of a nonlinear left-handed material and observe generation and propagation of temporal solitons in such materials. We demonstrate also that the nonlinear left-handed metamaterials can support both transverse electric– and transverse magnetic–polarized self-trapped localized beams, spatial electromagnetic solitons. Such solitons appear as single-hump and multihump beams, being either symmetric or antisymmetric, and they can exist because of the hysteresis-type magnetic nonlinearity and the effective domains of negative magnetic permeability.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Nonlinear Resonant Response
  5. 3. FDTD Simulations of Nonlinear Transmission
  6. 4. Electromagnetic Spatial Solitons
  7. 5. Conclusions
  8. Acknowledgments
  9. References

[2] Recent theoretical studies [Pendry et al., 1996, 1999; Markos and Soukoulis, 2002a, 2002b] and experimental results [Smith et al., 2000; Bayindir et al., 2002; Parazzoli et al., 2003] have shown the possibility of creating novel types of microstructured materials that demonstrate many intriguing properties such as negative refraction. In particular, the composite materials created by arrays of wires and split-ring resonators were shown to possess a negative real parts of the magnetic permeability and the dielectric permittivity for microwaves. These materials are often referred to as left-handed materials (LHMs) or materials with negative refraction. Properties of the left-handed materials were analyzed theoretically by Veselago [1967] a long time ago, but such materials were demonstrated experimentally only very recently. As was shown by Veselago [1967], the left-handed materials possess a number of peculiar properties, including negative refraction for interface scattering, inverse light pressure, reverse Doppler and Vavilov-Cherenkov effects, etc.

[3] So far, most of the properties of left-handed materials were studied in the linear regime of wave propagation when both magnetic permeability and dielectric permittivity of the material are assumed to be independent on the intensity of the electromagnetic field. However, the future efforts in creating tunable structures where the field intensity changes the transmission properties of the composite structure would require the study of nonlinear properties of such metamaterials, which may be quite unusual. In particular, the recently fabricated metamaterials are composed of a mesh of wires and split-ring resonators (SRRs). The wires provide negative dielectric permittivity, while SRRs give negative magnetic permeability. Metamaterials possess left-handed properties only in some finite frequency range, which is basically determined by the geometry of the structure. The possibility to control the effective parameters of the metamaterial using nonlinearity has recently been suggested by Zharov et al. [2003] and Lapine et al. [2003]. Importantly, the microscopic electric field in the left-handed structure can be much higher than the macroscopic electric field carried by the propagating wave. This provides a simple physical mechanism for enhancing nonlinear effects in left-handed materials.

[4] In this paper we present a brief overview of some nonlinear properties of left-handed metamaterials for the example of a lattice of SRRs and wires with a nonlinear dielectric. By means of the finite difference time domain (FDTD) simulations, we study the wave scattering by a slab of a nonlinear composite structure. We also discuss the structure of electromagnetic solitons supported by the nonlinear left-handed materials with hysteresis-type nonlinear response. We believe our findings may stimulate the future experiments in this field, as well as the studies of nonlinear effects in photonic crystals, where the phenomenon of negative refraction is analyzed now very intensively [Luo et al., 2002a, 2002b].

2. Nonlinear Resonant Response

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Nonlinear Resonant Response
  5. 3. FDTD Simulations of Nonlinear Transmission
  6. 4. Electromagnetic Spatial Solitons
  7. 5. Conclusions
  8. Acknowledgments
  9. References

[5] First, we extend the results of the original paper [Zharov et al., 2003] and consider a three-dimensional composite structure in the form of a cubic lattice of conducting wires and SRRs. The wires are placed at the edges of the cube, while SRRs are placed at its facets. We assume that the unit cell size dcell of the structure is much smaller than the wavelength of the propagating electromagnetic field and, for simplicity, we choose the single-ring geometry of a lattice of SRRs. The results obtained for this case are qualitatively similar to those obtained in more involved cases of double SRRs. This type of microstructured materials has recently been suggested and built in order to create left-handed metamaterials with negative refraction in the microwave region [Smith et al., 2000].

[6] The negative real part of the effective dielectric permittivity of such a composite structure appears because of the metallic wires whereas a negative sign of the magnetic permeability becomes possible because of the SRR lattice. As a result, these materials demonstrate the properties of negative refraction in a finite frequency range, that is, ω0 < ω < min(ωp, ωm), where ω0 is the eigenfrequency of the SRRs, ωm is the frequency of the longitudinal magnetic plasmon, ωp is the effective plasma frequency, and ω is the angular frequency of the propagating electromagnetic waves, (ℰ, ℋ) ∼ (E, H) exp(iωt). The SRR can be described as an effective LC oscillator [see, e.g., Gorkunov et al., 2002] with capacitance of the SRR gap, as well as an effective inductance and resistance.

[7] We assume that only the slits of the SRRs are filled with nonlinear dielectric with a permittivity that depends on the intensity of the electric field ∣E2 in a rather general form, εD = εD(∣E2). For the calculations presented below, we take the dependence that corresponds to the Kerr-type nonlinear response, εD = εl + α∣E2/Ec2, where εl is the linear part of the dielectric permittivity, Ec is a characteristic electric field strength, α = +1 for focusing nonlinearity and α = −1 for defocusing nonlinearity.

[8] The nonlinear response of the composite material comes from the lattice of resonators, since the SRR capacitance (and therefore the SRR eigenfrequency) depends on the strength of the local electric field in a narrow slot. The intensity of the local electric field in the SRR gap, Eg, depends on the electromotive force in the resonator loop, which is induced by the magnetic field. Therefore the effective magnetic permeability μeff depends on the macroscopic (average) magnetic field H, and this dependence can be found in the three-dimensional case similar to the two-dimensional geometry discussed by Zharov et al. [2003],

  • equation image

where

  • equation image

is the eigenfrequency of nonlinear oscillations, Γ = c2/4πσarw is the dumping coefficient, F = π2a3/2dcell3 [ln(8a/rw) − 7/4] is the filling factor, a is the SRR radius, rw is the radius of the SRR wire, σ is the conductivity of the wires, Eg is the strength of the electric field in the SRR slit, c is the speed of light. It is important to note that equation (1) has a simple physical interpretation: The resonant frequency of the artificial magnetic structure depends on the amplitude of the external magnetic field and, in turn, this leads to the intensity-dependent function μeff.

[9] Figures 1 and 2 summarize different types of the nonlinear magnetic properties of the composite, which are defined by the dimensionless frequency of the external field Ω = ω/ω0, for both focusing (Figures 1a, 1b, 2a, and 2b) and defocusing (Figures 1c, 1d, 2c and 2d) nonlinearity of the dielectric.

image

Figure 1. Real part of the effective magnetic permeability versus intensity of the magnetic field: (a) Ω > 1, α = 1; (b) Ω < 1, α = 1, (c) Ω > 1, α = −1; and (d) Ω < 1, α = −1. Black curves represent the lossless case (γ = 0), and green curves represent the lossy case (γ = 0.05). Dashed curves show unstable branches.

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image

Figure 2. Imaginary part of the effective magnetic permeability versus intensity of the magnetic field for γ = 0.05: (a) Ω > 1, α = 1; (b) Ω < 1, α = 1, (c) Ω > 1, α = −1; and (d) Ω < 1, α = −1. Dashed curves show unstable branches.

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[10] Because of high amplitude of the electric field in the SRR slit as well as resonant interaction of the electromagnetic field with the SRR lattice, nonlinear effects in such structures can be enhanced dramatically. Moreover, the critical fields for switching between the LH and RH states, shown in Figure 1 can be reduced to a desirable value by choosing the frequency close to the resonant frequency of SRRs. Even for a relatively large difference between the SRR eigenfrequency and the external frequency, as in Figure 1b where Ω = 0.8 (i.e., ω = 0.8ω0), the switching amplitude of the magnetic field is ∼0.03Ec. The characteristic values of the focusing nonlinearity can be estimated for some materials such as n-InSb for which Ec = 200 V/cm [Belyantsev et al., 1981]. As a result, the strength of the critical magnetic field is found as Hc1 ≈ 1.6 A/m. Strong defocusing properties for microwave frequencies are found in BaxSr1–xTiO3 [see Li et al., 2001, and references therein]. The critical nonlinear field of a thin film of this material is Ec = 4 × 104 V/cm, and the corresponding field of the transition from the LH to RH state (see Figure 1c) can be found as Hc ≈ 55.4 A/m.

[11] The unique possibility of strongly enhanced effective nonlinearities in left-handed metamaterials revealed here may lead to an essential revision of the concepts based on the linear theory, since the electromagnetic waves propagating in such materials always have a finite amplitude. At the same time, the engineering of nonlinear composite materials may open a number of their novel applications such as frequency multipliers, beam spatial spectrum transformers, switchers, limiters, etc.

3. FDTD Simulations of Nonlinear Transmission

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Nonlinear Resonant Response
  5. 3. FDTD Simulations of Nonlinear Transmission
  6. 4. Electromagnetic Spatial Solitons
  7. 5. Conclusions
  8. Acknowledgments
  9. References

[12] In order to verify the specific features of the left-handed metamaterials introduced by their nonlinear response, in this section we study the scattering of electromagnetic waves from the nonlinear metamaterial discussed above. In particular, we perform the FDTD numerical simulations of the plane wave interaction with a slab of LHM of a finite thickness. We use Maxwell's equations in the form

  • equation image
  • equation image

where 〈j〉 is the current density averaged over the period of the unit cell, and M is the magnetization of the metamaterial. We base our analysis on the microscopic model recently discussed by Shadrivov et al. [2004], and write the constitutive relations in the form

  • equation image
  • equation image

where Lw is the inductance of the wire per unit length, S is the effective cross section of a wire, S ≈ πrw2, for δ > rw, and S ≈ πδ(2rw − δ), for δ < rw, where δ = c/equation image is the skin layer thickness, IR is the current in SRR, nm is concentration of SRRs. The current in SRRs is governed by the equation

  • equation image

where L is inductance of the SRR, R is resistance of the SRR wire, U is the voltage on the SRR slit, and H′ is the acting (microscopic) magnetic field, which differs from the average (macroscopic) magnetic field. Voltage U at the slit of SRR is coupled to the current IR through the relation

  • equation image

with

  • equation image

where Uc is the characteristic nonlinear voltage, Uc = Ec × dg.

[13] The microscopic magnetic field H′ can be expressed in terms of M and B using the Lorenz-Lorentz relation [Born and Wolf, 2002], assuming that the number of SRRs in the volume λ3, where λ denotes the wavelength, is much greater than 1:

  • equation image

As a result, equations (2)–(6) form a closed system of coupled equations, and they can be solved numerically using, for example, the numerical FDTD method. We also notice that, by substituting the harmonic fields into these equations, we recover the expression for the magnetic permeability (1).

[14] Our goal is to study the temporal dynamics of the wave scattering by a finite slab of nonlinear metamaterial. For simplicity, we consider a one-dimensional problem that describes the interaction of the plane wave incident at the normal angle from air on a slab of metamaterial of a finite thickness. We consider two types of nonlinear effects: (1) nonlinearity-induced suppression of the wave transmission when initially transparent left-handed material becomes opaque with the growth of the input amplitude, and (2) nonlinearity-induced transparency when an opaque metamaterial becomes left-handed (and therefore transparent) with the growth of the input amplitude. The first case corresponds to the dependence of the effective magnetic permeability on the external field shown in Figures 1a and 2c, when initially negative magnetic permeability (we consider ε < 0 in all frequency range) becomes positive with the growth of the magnetic field intensity. The second case corresponds to the dependence of the magnetic permeability on the external field shown in Figure 1b.

[15] In all numerical simulations, we use linearly growing amplitude of the incident field within the first 50 periods, that becomes constant afterward. The slab thickness is selected as 1.3λ0 where λ0 is a free space wavelength. For the parameters we have chosen, the metamaterial is left handed in the linear regime for the frequency range from f1 = 5.787 GHz to f2 = 6.05 GHz.

[16] Our simulations show that for the incident wave with the frequency f0 = 5.9 GHz (i.e., inside the left-handed transmission band), electromagnetic field reaches a steady state independently of the sign of the nonlinearity. Both reflection and transmission coefficients in the stationary regime are shown in Figures 3 and 4 as functions of the incident field amplitude, for both defocusing and focusing nonlinear response of the dielectric infilling the SRR slits. In the linear regime, the effective parameters of the metamaterial at the frequency f0 are: ε = −1.33 – 0.01i and μ = −1.27 – 0.3i; this allows excellent impedance matching with surrounding air. The scattering results in a vanishing reflection coefficient for small incident intensities (see Figures 3 and 4).

image

Figure 3. Reflection (solid curve) and transmission (dashed curve) coefficients for a slab of nonlinear metamaterial versus the incident field intensity in a stationary regime for the case of defocusing nonlinearity (α = −1). Inset shows real (solid curve) and imaginary (dashed curve) parts of the magnetic permeability inside the slab.

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image

Figure 4. Reflection (solid curve) and transmission (dashed curve) coefficients for a slab of nonlinear metamaterial versus the incident field intensity in a stationary regime for the focusing nonlinearity (α = 1). Inset shows real (solid curve) and imaginary (dashed curve) parts of the magnetic permeability inside the slab.

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[17] Reflection and transmission coefficients are qualitatively different for two different types of infilling nonlinear dielectric. For the defocusing nonlinearity, the reflection coefficient varies from low to high values when the incident field exceeds some threshold value (see Figure 3). Such a sharp transition can be explained in terms of the hysteresis behavior of the magnetic permeability shown in Figure 1c. When the field amplitude in metamaterial becomes higher than the critical amplitude (shown by a dashed arrow in Figure 1c), magnetic permeability changes its sign, and the metamaterial becomes opaque. Our FDTD simulations show that for overcritical amplitudes of the incident field, the opaque region of positive magnetic permeability appears inside the slab (see the inset in Figure 3). The magnetic permeability experiences an abrupt change at the boundary between the transparent and opaque regions. The dependencies shown in Figure 3 are obtained for the case when the incident field grows from zero to a steady state value. However, taking different temporal behavior of the incident wave, for example, increasing the amplitude above the threshold value and then decreasing it to the steady state, one can get different values of the stationary reflection and transmission coefficients, and different distributions of the magnetic permeability inside the metamaterial slab. Such properties of the nonlinear metamaterial slab are consistent with the predicted multivalued dependence of the magnetic permeability on the amplitude of the magnetic field.

[18] In the case of focusing nonlinearity (see Figure 4), the dependence of the reflection and transmission coefficients on the amplitude of the incident field is smooth. This effect originates from a gradual detuning from the impedance matching condition, and, for higher powers, from the appearance of an opaque layer (see the inset in Figure 4) with a positive value of the magnetic permeability that is a continuous function of the coordinate inside the slab.

[19] Now we consider another interesting case when initially opaque metamaterial becomes transparent with the growth of the incident field amplitude. We take the frequency of the incident field to be f0 = 5.67 GHz, so that magnetic permeability is positive in the linear regime and the metamaterial is opaque. In the case of self-focusing nonlinear response (α = 1), it is possible to switch the material properties to the regime with negative magnetic permeability (see Figure 1b) making the material slab left handed and therefore transparent. Moreover, one can expect the formation of self-focused localized states inside the composite, the effect which was previously discussed for the interaction of the intense electromagnetic waves with overdense plasma [Zauer and Gorbunov, 1977; Zharov and Kotov, 1984; Kochetov and Feigin, 1988]. Figure 5a shows the temporal evolution of the incident and reflected wave intensities for small input intensities, this case corresponds to the linear regime. The reflection coefficient reaches a steady state after approximately 100 periods. The spatial distribution of the electric and magnetic fields at the end of simulation time is shown in Figures 5b and 5c, respectively.

image

Figure 5. (a) Reflected (solid curve) and incident (dashed curve) wave intensity versus time for small amplitudes of the incident wave (i.e., in the linear regime). (b, c) Distribution of the magnetic and electric fields, respectively, at the end of simulation time; the metamaterial is shaded.

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[20] In a weakly nonlinear overcritical regime (see Figure 6), the intensity of the reflected beam decreases approaching a steady state. In this case we observe the formation of a localized state inside the metamaterial slab and near the interface, as can be seen more distinctly in Figure 6c. This effect give an additional contribution to the absorption of the electromagnetic energy, thus leading to a decay of the value of the reflection coefficient.

image

Figure 6. Same as Figure 5 but in the regime of critical amplitude of the incident wave.

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[21] In a strongly nonlinear overcritical regime, we observe the effect of the dynamical self-modulation of the reflected electromagnetic wave that results from the periodic generation of the self-localized states inside the metamaterial (see Figure 7). Such localized states resemble temporal solitons, which transfer the energy away from the interface. Figure 7c shows an example when two localized states enter the metamaterial. These localized states appear on the jumps of the magnetic permeability and, as a result, we observe a change of the sign of the electric field derivative at the maximum of the soliton intensity, and subsequent appearance of transparent regions in the metamaterial. Unlike all previous cases, the field structure in this regime do not reach any steady state for high enough intensities of the incident field.

image

Figure 7. Same as Figure 5 but in the regime of overcritical amplitude of the incident wave.

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4. Electromagnetic Spatial Solitons

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Nonlinear Resonant Response
  5. 3. FDTD Simulations of Nonlinear Transmission
  6. 4. Electromagnetic Spatial Solitons
  7. 5. Conclusions
  8. Acknowledgments
  9. References

[22] Similar to other nonlinear media [Kivshar and Agrawal, 2003], nonlinear left-handed composite materials can support self-trapped electromagnetic waves in the form of spatial solitons. Such solitons possess interesting properties because they exist in materials with a hysteresis-type (multistable) nonlinear magnetic response. Below, we describe novel and unique types of single-hump and multihump (symmetric, antisymmetric, or even asymmetric) backward wave spatial electromagnetic solitons supported by the nonlinear magnetic permeability.

[23] Spatially localized transverse magnetic–polarized waves that are described by one component of the magnetic field and two components of the electric field. Monochromatic stationary waves with the magnetic field component H = Hy propagating along the z axis and homogeneous in the y direction, [∼exp (iωtikz)], are described by the dimensionless nonlinear Helmholtz equation

  • equation image

where γ = kc/ω is a wave number, x = x′ω/c is the dimensionless coordinate, and x′ is the dimensional coordinate. Different types of localized solutions of equation (7) can be analyzed on the phase plane (H, dH/dx) [see, e.g., Gil'denburg et al., 1983]. First, we find the equilibrium points: the point (0, 0) existing for all parameters, and the point (0, H1), where H1 is found as a solution of the equation

  • equation image

Below the threshold, that is, for γ < γtr, where γtr2 = ε [1 + FΩ2/(1 − Ω2)], the only equilibrium state (0, 0) is a saddle point, and therefore no finite amplitude or localized waves can exist. Above the threshold value, that is, for γ > γtr, the phase plane has three equilibrium points, and a separatrix curve corresponds to a soliton solution.

[24] In the vicinity of the equilibrium state (0, 0), linear solutions of equation (7) describe either exponentially growing or exponentially decaying modes. The equilibrium state (0, H1) describes a finite amplitude wave mode of the transverse electromagnetic field. In the region of multistability, the type of the phase trajectories is defined by the corresponding branch of the multivalued magnetic permeability. Correspondingly, different types of the spatial solitons appear when the phase trajectories correspond to the different branches of the nonlinear magnetic permeability.

[25] The fundamental soliton is described by the separatrix trajectory on the plane (H, dH/dx) that starts at the point (0, 0), goes around the center point (0, H1), and then returns back; the corresponding soliton profile is shown in Figure 8a. More complex solitons are formed when the magnetic permeability becomes multivalued and is described by several branches. Then, soliton solutions are obtained by switching between the separatrix trajectories corresponding to different (upper and lower) branches of magnetic permeability. Continuity of the tangential components of the electric and magnetic fields at the boundaries of the domains with different values of magnetic permeability implies that both H and dH/dx should be continuous. As a result, the transitions between different phase trajectories should be continuous.

image

Figure 8. Examples of different types of solitons: (a) fundamental soliton, (b) soliton with one domain of negative magnetic permeability (shaded), (c) soliton with one domain of positive magnetic permeability (shaded), and (d) soliton with two different domains (shaded). Insets in Figures 8b and 8c show the magnified regions of the steep change of the magnetic field.

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[26] Figures 8b and 8c show several examples of the more complex solitons corresponding to a single jump to the lower branch of μeff(H) (see Figure 1c) and to the upper branch of μeff(H). The insets show the magnified domains of a steep change of the magnetic field. Both the magnetic field and its derivative, proportional to the tangential component of the electric field, are continuous. The shaded areas show the effective domains where the value of magnetic permeability changes. Figure 8d shows an example of more complicated multihump soliton which includes two domains of the effective magnetic permeability, one described by the lower branch, and the other one by the upper branch. In a similar way, we can find more complicated solitons with different number of domains of the effective magnetic permeability.

[27] We note that some of the phase trajectories have discontinuity of the derivative at H = 0 caused by infinite values of the magnetic permeability at the corresponding branch of μeff(H). Such a nonphysical effect is an artifact of the lossless model of a left-handed nonlinear composite considered here for the analysis of the soliton solutions. In more realistic models that include losses, the region of multistability does not extend to the point H = 0, and in this limit the magnetic permeability remains a single-valued function of the magnetic field [Zharov et al., 2003].

[28] For such a multivalued nonlinear magnetic response, the domains with different values of the magnetic permeability “excited” by the spatial soliton can be viewed as effective induced left-handed waveguides which make possible the existence of single-hump and multihump soliton structures. Because of the existence of such domains, the solitons can be not only symmetric, but also antisymmetric and even asymmetric. Formally, the size of an effective domain can be much smaller than the wavelength, and therefore there exists an applicability limit for the obtained results to describe nonlinear waves in realistic composite structures.

[29] When the infilling dielectric of the structure displays self-focusing nonlinear response, we have Ω < 1, and in such a system we can find dark solitons, that is, localized dips on the finite amplitude background wave [Kivshar and Agrawal, 2003]. Similar to bright solitons, there exist both fundamental dark solitons and dark solitons with domains of different values of magnetic permeability. For self-defocusing nonlinearity and Ω < 1, magnetic permeability is a single-valued function, and such a nonlinear response can support dark solitons as well, whereas for self-focusing dielectric, we have Ω > 1 and no dark solitons can exist.

5. Conclusions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Nonlinear Resonant Response
  5. 3. FDTD Simulations of Nonlinear Transmission
  6. 4. Electromagnetic Spatial Solitons
  7. 5. Conclusions
  8. Acknowledgments
  9. References

[30] We have discussed novel properties of left-handed metamaterials associated with their nonlinear resonant response. For the case of harmonic fields, we have calculated the effective magnetic permeability of microstructured materials consisting of rods and split-ring resonators, and predicted the hysteresis-like dependence of the nonlinear magnetic permeability as a function of the applied magnetic field. Using the finite difference time domain numerical simulations, we have studied the temporal dynamics of the wave reflection from a slab of nonlinear metamaterial that is found to be consistent with our theory. Finally, we have predicted the existence of electromagnetic spatial solitons supported by the hysteresis-type nonlinear magnetic permeability of a left-handed material.

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Nonlinear Resonant Response
  5. 3. FDTD Simulations of Nonlinear Transmission
  6. 4. Electromagnetic Spatial Solitons
  7. 5. Conclusions
  8. Acknowledgments
  9. References

[31] This work has been supported by the Australian Research Council and the Australian National University. A.A.Z. acknowledges support from the Russian Foundation for Basic Research (grant 05-02-16357).

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Nonlinear Resonant Response
  5. 3. FDTD Simulations of Nonlinear Transmission
  6. 4. Electromagnetic Spatial Solitons
  7. 5. Conclusions
  8. Acknowledgments
  9. References
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