Bright spatial solitons in controlled negative phase metamaterials

Optics Communications 283 (2010) 1585–1597

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Invited paper

Bright spatial solitons in controlled negative phase metamaterials A.D. Boardman *, R.C. Mitchell-Thomas, N.J. King, Y.G. Rapoport Joule Physics Laboratory, Institute for Materials Research, University of Salford, Greater Manchester, United Kingdom

a r t i c l e

i n f o

Article history: Received 23 July 2009 Received in revised form 10 September 2009 Accepted 11 September 2009

Keywords: Soliton Magneto-optic Nonlinear Metamaterial Diffraction-management Higher-order linear diffraction

a b s t r a c t A fundamental discussion of linearly polarised bright spatial soliton beams that are localised in a planar waveguide is developed in a special way that leads to the introduction of nonlinear diffraction. It is emphasised, throughout, that this kind of diffraction dominates influences from non-paraxiality and typical quintic contributions to the nonlinearity. A major discussion is given that is based upon double-negative metamaterials and makes contact with previous literature. Both homogeneous and inhomogeneous diffraction-managed, planar waveguides are investigated with a view to examining the behaviour of narrow beam production and propagation. It is also shown that a Voigt configuration of an externally applied magnetic field can be used to create significant magnetooptic control over spatial soliton propagation in asymmetric waveguides. Both this type of control and diffraction-management lie at the heart of the numerical simulations given here. In all of these cases it is shown that nonlinear diffraction has a very important influence and will create an impact upon future applications. Ó 2010 Elsevier B.V. All rights reserved.

1. Introduction Solitons, in hydrodynamics and electromagnetism, have been studied for a long time [1,2] and have found particular importance in the optical domain [3], especially in connection with optical fibres [4]. It is therefore very important to investigate the extent to which new materials are capable of sustaining various types of solitons. In this context, metamaterials may be suitable vehicles for soliton excitations, and this paper sets out some basic principles in this area, and introduces some new ideas that could lead to applications. The soliton family is very large, so it is necessary to restrict attention to particular members and one of them comes under the heading of spatial solitons [5–11]. It is these that will be addressed here, and they are beams of electromagnetic energy that rely upon balancing chirps associated with diffraction and nonlinearity to retain their shape. This property is in contrast to temporal solitons that are pulses which rely upon balancing phase changes [4] across their width, which arise from material dispersion and nonlinearity. Spatial solitons, like their temporal counterparts, are stable if they are solutions of what is known as the one-dimensional cubic nonlinear Schrödinger equation. Hence, although a beam of electromagnetic energy in a bulk medium has two free transverse-dimensional degrees of freedom, it can, in principle, balance diffraction with nonlinearity, but the balance is unstable. It was shown, some time ago, however, that placing a beam in a * Corresponding author. Tel.: +44 777 616 2025. E-mail address: [email protected] (A.D. Boardman). 0030-4018/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2009.09.024

planar waveguide produces stability in an elegant fashion, and this is the basic model to be adopted here [12]. Within a planar waveguide, a stable soliton will be created by permitting the beam to diffract in the plane of the guide and the role of any diffractionmanagement that is present, naturally, or artificially, created will be important. Indeed, for positive phase materials, diffractionmanagement, and also managing dispersion, has been investigated already [13], not only for optical fibres, but also for spatial solitons [14], especially through the utilization of waveguide arrays [15]. The basic nonlinear Schrödinger equation is often an adequate model for the behaviour of electromagnetic beams but it is often the case that important additions to this core equation have to be included. One example is the appearance of non-paraxial terms when the slowly-varying amplitude approximation is partially relaxed. In this paper, it will be shown, however, that nonlinearly-induced diffraction [3,16,17] is a vital addition to the nonlinear Schrodinger equation. This is because the latter dominates as the beams become very narrow and can be more important than the non-paraxial terms and quintic nonlinearity in preventing beam collapse at high powers. The interplay between nonlinear diffraction and the possibility of managing the usual diffraction [18] is fascinating, and is something that will be discussed below through the agency of the negative phase behaviour that can be associated with metamaterials [19]. Even though the core nonlinear Schrodinger equation is modified to take into account other effects, the solutions will still be referred to as ‘solitons’. As previously pointed out [2], in the literature, this is a convenient and common practice when dealing with excitations that look and behave like solitons throughout their range of applicability.

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The study of solitons in a nonlinear negative metamaterial has resulted in an elegant formulation of a generalised nonlinear Schrödinger equation [20] in which it is emphasised that by using the dispersion brought in by the relative dielectric permittivity and the relative permeability, a lot of new possibilities can be anticipated. For example, the sign of the self-steepening parameter can be changed through frequency management of the operational windows. This point has been vigorously exploited in recent studies of modulation instability and short pulse propagation [21–24]. A study of the regions close to the band edges of the relative dielectric permittivity and the relative permeability will also reveal whether slow solitons are possible. This has been discussed recently [25] together with the intensities needed to launch such interesting beams. In contrast, the work that will now unfold will not address dispersion but diffraction. It will lead to dimensionless equations that deploy coefficients calculated for a purely monochromatic beam. The metamaterial properties, therefore, will emerge now as an influence upon the nonlinearly-induced diffraction. A fundamental derivation of the nonlinear Schrödinger equation will be presented that includes diffraction-management and nonlinear diffraction. It will not include loss, however, because as recent literature shows, it is possible to diminish the loss [26– 29] to a very small influence, whilst retaining the kind of doublenegative behaviour that is such an attractive feature of negative phase metamaterials. It is interesting that there is a complete analogy between the nonlinear Schrödinger work in the literature, and the study of unstable waves on deep oceans. Such instability in the hydrodynamic area was met with considerable scepticism, but it has been shown to be correct that the conditions for modulation instability are indeed the conditions for the creation of solitons. This applies to both spatial and temporal solitons [30,31].

2. Beam evolution in nonlinear planar waveguide 2.1. Basic template Here, a basic Schrödinger equation template is developed, to which can be added nonlinear diffraction and special metamaterial properties, as the discussion develops in the other sections. The template emerges in a form that is suitable for the propagation of an optical beam located in a planar waveguide, as sketched in Fig. 1. The equation generically accounts for non-paraxiality, a simple Kerr-nonlinearity and the impact of the spatial Fourier components of the beam upon the diffraction taking place in the plane of the guide. The spatial soliton that is established in the planar guide is stable because the diffraction along the y-axis is frozen and replaced by guiding. After establishing this basic Schrödinger

equation, nonlinear diffraction will be discussed in a separate physical argument that leads to an elegant modification of the generic equation. Finally, an equation, applicable to a metamaterial is developed. For the moment, then, only nonmagnetic material for which the permeability is l = 1 will be considered. If E is the electric field vector of an electromagnetic beam travelling through a polarisable medium, then, from Maxwell’s equations

r2 E  rðr  EÞ ¼ l0

@2D @ 2 PNL þ l 0 @t 2 @t 2

ð1:1Þ

where D is the linear displacement vector, PNL is the nonlinear electric polarisation of the dielectric and l0 is the permeability of free space. The standard assumption in the literature is to set div (E) = 0. This will be challenged below but, at this stage, the basic starting point is

e¼ r2 E

x20 e x20 e e E  2 De E 2 c

ð1:2Þ

c





e  eix0 t is used, and it is assumed that the e ix0 t þ E where E ¼ 12 Ee beam is operating at a fixed frequency x0, e = n2 is the linear dielectric constant, n is the linear refractive index, l0e0 = 1/c2, where e0 is the permittivity of free space, c is the velocity of light in vacuo and e 2 3 ð3Þ e 2 e2 De ¼ eð3Þ NL j Ej ¼ 4 vxxxx j Ej ¼ aj Ej and is the nonlinear addition to the ð3Þ linear dielectric constant. vxxxx is the appropriate term from the nonlinear susceptibility tensor for an isotropic material. If the nonlinearity is weak, then the modal field is, in practice, unaffected by the nonlinearity. If, for example, a TE-mode is adopted, the electric e ¼ ðEx ; 0; 0Þ which can be defined as field is then E

Z  Ex ¼ CAðyÞ Eðkx ; zÞeikx x dkx eikz0 z

ð1:3Þ

where C is a constant, E(kx, z) is a Fourier component that is a function of z, A(y) is taken to be independent of z and approximately satisfies the modal field equation of the guide. The integration defines a packet of waves with a wave number distribution kx. The latter defines the spatial extent of the beam. kz0 is the wave number along the z-axis and, to this order, is unperturbed by the presence of the kx wave numbers. The distribution (1.3), after substitution into (1.2), leads to the equations

1 @ 2 A x20 2 x2 þ n A þ 20 DeA A @y2 c2 c

! 2 ¼k

1 @E @ 2 E 2 2  kz0 E  kx E 2ikz0 þ E @z @z2

ð1:4Þ

! 2 ¼ k

ð1:5Þ

where the method of separation of variables is used, with the aid of 2 , which is selected here to be the separation quantity k 2

2 ¼ k2 þ k2 þ x0 DNL k z x c2 Y

2 is interpreted as the nonlinear eigenvalue that determines In fact, k the nonlinear shift from the linear value. The shift is DNL. This can be found in terms of the modal fields A(y) in the following way. Eq. (1.4), after substituting (1.6), and then multiplying by the complex amplitude, A*, followed by an integration over y, yields

Cladding

Guide

H

ð1:6Þ

E X

Substrate Z Fig. 1. Sketch of a typical TE-polarised electromagnetic beam trapped in a planar waveguide of infinite extent in the x direction. This is an illustration of a (1 + 1) spatial soliton with diffraction in the x-direction and propagation in the z direction. The beam carries the electromagnetic field components (Ex, Hy).

R R 2 DejAj2 dy C ajAj2 jEj2 jAj2 dy DNL ¼ R 2 ¼ ¼ a0 jEj2 R 2 jAj dy jAj dy

ð1:7Þ

The normalisation constant C is arbitrary and is selected here to e0

ncjEj2 . This step is based upon normalising the intensity by defining C2 ¼ R 12 . Hence, a0 ¼ wa where

make the effective intensity

2

jAj dy

the effective width of the planar guide is now

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R w¼

R

2 jAj dy

2 ð1:8Þ

4 jAj dy

Since the role of the modal field is often not included in the literature, Eq. (1.8) has been developed to show how it enters the slowlyvarying amplitude equation. It is clear that for a spatial soliton in a planar waveguide it is an effective width, w, that appears. Even if a higher-order nonlinearity is considered, the effective width will still be approximately (1.8). From this analysis, it can be appreciated @2 that a @y 2 term will not appear in the final nonlinear Schrödinger equation and, in practice, it means that the role of the planar waveguide containing the spatial soliton effectively behaves as a (1 + 1) system by ‘‘freezing” the y-coordinate and permitting diffraction only in the x-direction. It will be assumed in the rest of the paper that this feature is present in all the normalised equations, without specifically discussing it. kz  kz(kx), i.e. kz is the perturbed wave number associated with the z-axis. Hence, the equation for E, after using Eq. (1.6) is

2ikz0

 @E @ 2 E  2 x2 2 þ 2 þ kz  kz0 E þ 20 DNL E ¼ 0 @z @z c

ð1:9Þ

where DNL does not actually need to be restricted to third-order nonlinearity, because De, defined earlier, could include a fifth-order nonð5Þ xxxxxx

5v

2

2

linearity eNL jEj4 E ¼ 8 jEj4 E, and in which ðkz  kz0 Þ ’ 2kz0 ðkz  kz0 Þ, and (kz  kz0) can be expressed as a Taylor series in kx, i.e. ð5Þ



@kz kz  kz0 þ @kx



1 @ 4 kz þ 24 @k4x 2

1 @ 2 kz kx þ 2 @k2x kx ¼0 ! 4 kx

! 2 kx kx ¼0

1 @ 3 kz þ 6 @k3x

þ ...

! 3

kx kx ¼0

ð1:10Þ

kx ¼0

2

where kx þ kz is a constant, so that the leading terms are the third and fifth, and they translate into x-derivatives in the following way.

1 2 1 @2 kx ) 2kz0 2kz0 @ 2 x 1 4 1 @4  3 kx )  3 8kz0 8kz0 @x4



ð1:11Þ ð1:12Þ

This development leads to the standard diffraction, and a higherorder linear diffraction with a form that has been reported before [32,33]. At this stage, therefore, the basic envelope equation for a nonmagnetic material is 2

2ikz0 þ

2

4

2 0 2 c

@E @ E @ E 1 @ E 3x þ  þ þ @z @z2 @x2 4k2z0 @x4 4 5 x20 ð5Þ v jEj4 E ¼ 0 8 c2 xxxxxx

ð3Þ vxxxx jEj2 E

ð1:13Þ

This equation includes the possibility of higher-order linear diffraction, but because such a term ranks with fifth-order nonlinearity, the latter is also included for completeness. Even though Eq. (1.13) is still in dimensional form, it should be recognised that the nonlinear coefficients are modified by the effective width as demonstrated earlier. In practice, the same effective width can be used with sufficient accuracy for both the nonlinear terms. In addition to all of these effects, conditions will arise under which nonlinear diffraction [16,17] will play an vital role in the envelope equation. This possibility is usually eliminated, in the lit  e ¼ 0, where E e is the electric erature, by the assumption that div E field vector. In fact, this condition is broken down by the contribution of the nonlinear electric polarisation to the displacement vector and leads to a significant influence, as the beams become narrow. Additionally, a metamaterial may exhibit nonlinear mag-

netic polarisation, in which case the common assumption that   e ¼ 0, where H e is the magnetic field vector, also breaks div H down, leading once again to nonlinear diffraction. These possibilities lead to the conclusion that the behaviour of TE-polarised beams will differ from that of TM-polarised beams in negative phase metamaterials that are able to exhibit both electric and magnetic nonlinearity. This point will be taken up, in detail, later when non-reciprocity created by a magnetooptic influence is investigated. In order to demonstrate, clearly just how nonlinear diffraction can appear, the next section considers only some basic arguments for positive phase media. The conclusions will be vectored into the nonlinear Schrödinger equation for metamaterials in a later section. 2.2. Nonlinearly-induced diffraction In order to establish the basic principle of nonlinear diffraction, consider a non-dispersive positive phase medium, which sustains an electric field vector E that is a solution of the wave equation

r2 E  rðr  EÞ ¼ l0

@2D @ 2 PNL þ l0 2 @t 2 @t

ð2:1Þ

where D is the linear displacement vector, PNL is the nonlinear electric polarisation of the dielectric and l0 is the permeability of free space. As an example, assume that a TE-polarised beam is created in the planar guide sketched in Fig. 1, with an electric field component Ex and that the electric field component along the z-axis is jEz j  jExj. Assume also that the beam is monochromatic with an angular frequency x = x0, defined through the definitions, e ix0 t þ E e  eix0 t Þ, e NL eix0 t þ P e NL eix0 t Þ, e PNL ¼ 12 ð P D ¼ 12 ð D E ¼ 12 ð Ee ix0 t  ix0 t e þ D e Þ. Once again, for a weakly nonlinear guide, the e modal field of the waveguide in Fig. 1, is both linear and stationary, and will determine an effective guide-width. The way in which this can be done has been fully reviewed above. As stated earlier, the ycoordinate is effectively frozen with diffraction only in the x-direction. This means that the x-component of equation (2.1) has the form [16]

@ 2 Ex x20 @ 2 Ex 1 @ 2 PNL x  2 eðx0 ÞEx þ 2 ¼ x20 l0 PNL x  2 @z c @x e0 eðx0 Þ @x2 

1 @ 2 P NL z e0 eðx0 Þ @x@z

ð2:2Þ

Eq. (2.2) can then be processed by assuming a fast variation eikz0 z and introducing slowly-varying complex amplitudes that are only functions of x and z. i.e. the variables can be transformed as follows NL NL NL ikz0 z , where the ðEx ; PNL x ; P z Þ ) ðEðx; z; x0 Þ; P x ðx; z; x0 Þ; P z ðx; z; x0 ÞÞe explicit dependence upon the x and z coordinates, and the frequency are shown for clarity, but will not be explicitly shown in the development below. The longitudinal field component Ez is an order of magnitude smaller than the transverse component Ex, so that only PNL x will be used now. In this ‘scalar model’ the x-component of the nonlinear polarisation is, for a Kerr medium,

PNL x ¼

3 ð3Þ v e0 jEj2 E 4

ð2:3Þ

where v(3) is the third-order nonlinear coefficient. Hence, the modified nonlinear Schrödinger equation for which the final term models nonlinear diffraction is

@E 1 @ 2 E 3x2 vð3Þ 2 1 @2E 1 @4E þ  3 þ jEj E þ 2 2 2 4 @z kz0 @x kz0 @z 4c kz0 4kz0 @x 5x2 3vð3Þ @2  2  4 þ 2 0 vð5Þ jEj E ¼ 0 xxxxxx jEj E þ 8c kz0 4eðx0 Þkz0 @x2

2i

ð2:4Þ

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In arriving at (2.4), the role of the modal field is to create an effective waveguide width, the possibility of quintic nonlinearity has been included so that the role of nonlinear diffraction can be clarified, as well as the possibility of higher-order diffraction. The effectiveness of nonlinear diffraction, in competition with quintic nonlinearity, non-paraxiality and higher-order diffraction will be assessed later on. Its role within a negative phase metamaterial will also soon be made clear. What can be said at this stage, is that if only the nonlinear diffraction appears in the modified nonlinear Schrödinger equation, then its influence upon the control and formation of narrow beams will be strong, and this point will be checked in the simulations given below. For low-power beams, the nonlinear diffraction is not going to compete with the main dif@2 fraction term. As the beams become narrower, the effect of the @x 2 operator becomes stronger, and if the power in the beam becomes greater, then it should be expected that the nonlinear diffraction will begin to have an impact. In addition, the formulation of (2.4) can be readily extended to include higher-order nonlinearity by simply adding a quintic term, or it can be generalised to include the possibility of nonlinear saturation. Up to now the discussion has centred upon electric nonlinearity, but metamaterials may also display a magnetic nonlinearity. e – 0, thus creating a This type of nonlinearity will cause div ð HÞ nonlinear diffraction contribution determined by the nonlinear magnetic properties. For metamaterials, therefore, as will be appreciated later on, the beam polarisation and the dominance e ¼ 0 of one, or other, of the magnetic and electric nonlineardiv ð EÞ ity will control coupling to non-reciprocal magnetooptic behaviour. At this stage, no specific introduction of metamaterial behaviour has been effected, because the emphasis has been on accounting for the contributing influences upon soliton dynamics. It is now an appropriate time to introduce metamaterial behaviour. This will be done in the next section, in which attention will be drawn to the question of whether the electric and magnetic behaviour of a magnetic metamaterial is in irreducible coupled form [34], or whether it is reducible to uncoupled equations. 2.3. Envelope equation for a metamaterial For metamaterials, an interesting formulation has been given that uses a general magnetic nonlinearity that goes beyond a formulation involving Kerr [third-order] nonlinearity [35]. In terms of power available, a Kerr approximation is more probable with a possible extension into the weakly quintic regime. It will be assumed here, therefore, that the available nonlinearity is Kerr-like in both the dielectric and magnetic polarisations. For this situation, a conclusion has been drawn that the magnetic and electric field components are solutions of irreducible coupled equations with new types of solitons as solutions [34]. This outcome that the equations are irreducible is inconsistent, and arises because of the retention of corrections from an order that has already been neglected. This is supported by some recent work [26] and some very interesting work, on slow solitons [25]. An explicit derivation of the nonlinear Schrödinger equations, appropriate to a metamaterial with both nonlinear permittivity and permeability, will now be given. Since the major point to be made here concerns the formulation of envelope equations for a metamaterial that are either coupled or uncoupled, the roles of higher-order linear diffraction and nonlinear diffraction will be added in after the principle points have been made. The starting point is to consider a metamaterial capable of sustaining both nonlinear electric and magnetic polarisations. In this case, the frequency Fourier transforms of the constitutive relations are

e ðr; xÞ ¼ e0 eðxÞ E e ðr; xÞ þ P e NL ðr; xÞ D

ð2:5Þ

e ðr; xÞ þ l f e ðr; xÞ ¼ l lðxÞ H B 0 0 M NL ðr; xÞ

ð2:6Þ

where x is an angular frequency, r is a position vector, and the Fourier transforms of the displacement vector, the magnetic flux density vector, the electric field, the magnetic field, the dielectric nonlinear polarisation and the nonlinear magnetization are, respece NL . Any possible bianisotropy, anisotropy, or e B, e E, e H, e P e NL , M tively, D, spatial dispersion, is neglected. All the nonlinearity is accounted for e NL and f M NL . in the polarisations, P Using the Fourier transforms defined in (2.5) and (2.6), the required Maxwell’s equations are Z þ1 e ðr; xÞexp ðixtÞdx  l @ MNL ðr;tÞ ð2:7Þ r  Eðr;tÞ ¼ ixl0 lðxÞ H 0 @t 1 Z þ1 @ e r  Hðr;tÞ ¼  ixe0 eðxÞ E ðr; xÞexp ðixtÞdx þ PNL ðr;tÞ ð2:8Þ @t 1 Spatial solitons in the system sketched in Fig. 1 will undergo diffraction in the x-direction, but will operate at a fixed frequency, x0. Hence, after the transformations

1 Eðr; tÞ exp ðix0 t Þ þ c:c:; 2 1 Hðr; tÞ ) Hðr; tÞ exp ðix0 t Þ þ c:c: 2 1 PNL ðr; t Þ ) PNL ðr; t Þ exp ðix0 t Þ þ c:c:; 2 1 MNL ðr; t Þ ) MNL ðr; t Þ exp ðix0 tÞ þ c:c: 2

Eðr; tÞ )

and making the assumptions following equations emerge

r2 E  rðr  EÞ þ k20 ðx0 ÞE þ

x20 c2

r2 H  rðr  HÞ þ k20 ðx0 ÞH þ 2

x20 c2

lðx0 Þ

2 0 2 c

x

@MNL @t

 x0 MNL and PNL

e0

ð2:9Þ

@PNL @t

 x0 PNL , the

þ ix0 l0 r  MNL ¼ 0

eðx0 ÞMNL  ix0 r  PNL ¼ 0

ð2:10Þ ð2:11Þ

where k0 ðx0 Þ ¼ eðx0 Þlðx0 Þ. At this stage, the equations do appear to be coupled because of the curl of the magnetic and electric polarisations being positioned, respectively in Eqs. (2.10) and (2.11), but note that the terms that will produce nonlinear diffraction involve either r  E, or r  H, and these would not contribute to the coupling of these equations. Therefore, the question of coupling will be addressed without these terms being present. This logic also applies to non-paraxiality and higher-order diffraction. Hence, they will also be omitted for now, but will be added later to create a final form of the nonlinear Schrödinger equation that will be used for the numerical simulations. The notation can be simplified further now by adopting the definitions k  k0(x0), e  e(x0), l  l(x0). Furthermore, the development can be made more specific by considering a TE-polar^Hy , where x ^ and y ^ are unit ^ Ex and H ¼ y ised beam for which E ¼ x vectors. If the nonlinear polarisations are generated by Kerr-like reð3Þ ð3Þ ð3Þ sponses, then PNL ¼ e0 eNL jEx j2 Ex and MNL ¼ lNL jHy j2 Hy where eNL ð3Þ and lNL are, respectively, the electric and magnetic cubic nonlinearity coefficients. The latter is not a serious restriction because a saturable medium can easily be modelled by adding quintic terms to enhance the jExj2 and jHyj2 assumptions or by using suitably descriptive functions. For this (1 + 1) system, consisting of a planar waveguide supporting a beam, the y-coordinate will be assumed to be frozen out and only diffraction along x will be present. For the moment, however, the modal fields will not be explicitly introduced because the y-derivatives are suppressed. In practice, the freezing of the ycoordinate means that the modal fields are constant across the waveguide.

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The fast spatial variation, eikz0 z , can be extracted, now, by setting Ex ) Ex ðx; zÞeikz0 z , Hy ) Hy ðx; zÞeikz0 z and thus redefining Ex and Hy by the slowly-varying functions, with respect to z, Ex(x,z) and Hy (x, z). The x-component of Eq. (2.11), is

 0 0

r  H ¼ 

0 AH

0

1

AE  ikz0  B C ¼ i x e e 0 A; @ 0 0 0  0

 jAH =AE j2 ¼ ðe20 jej2 Þ=ðl0 jlje0 jejÞ ¼ 2ikz0

h   i @Ex @ 2 Ex x20 ð3Þ 2 ð3Þ ^ x ^¼0 þ le jEx j2 Ex þ ix0 l0 lNL r  Hy  Hy y þ @z @x2 c2 NL ð2:12Þ

^ component of the curl operator can be partially processed The x with standard vector identities, and then neglecting spatial derivatives of jHj2, on the grounds that they are of higher-order than required by this third-order equation. The next step is to make r  ðHy y^ Þ  ix0 e0 eðx0 ÞEx x^, after omitting the nonlinear polarisation because it yields a higher-order contribution. Processing Eqs. (2.10) and (2.11) in this manner leads to the following apparently coupled envelope equations for a metamaterial

 2 i @Ex @ 2 Ex x20 h ð3Þ ð3Þ þ 2 þ 2 leNL jEx j2 þ elNL Hy  Ex ¼ 0 @z @x c  2 i @Hy @ 2 Hy x20 h ð3Þ ð3Þ 2ikz0 þ 2 leNL jEx j2 þ elNL Hy  Hy ¼ 0 þ @z @x2 c

2ikz0

ð2:13Þ ð2:14Þ

ð3Þ

ð3Þ

where it should be noted that the nonlinear coefficients eNL and lNL can be positive or negative, and that a sign has not been attached to the wave number kz0. Eqs. (2.13) and (2.14) appear to support the conclusion that the electric and magnetic fields are coupled and that new forms of Ex  Hy coupled solitons can be created [34]. However, this conclusion is unsustainable, and Eqs. (2.13) and (2.14) are reducible. The arguments to support this will now be framed in terms of the properties of the slowly-varying functions Ex(x, z) and Hy(x,z) and their connection to the modal fields. The latter are discussed in the early part of the paper, in which the modal field A(y) was highlighted and it was vigorously emphasised that the electric and magnetic fields are assumed to behave as the product of a slowly-varying complex amplitude, a modal field variation arising from the guide containing the spatial soliton and a fast variation, eikz0 z that models the propagation down the guide axis. This is a standard approach, widely adopted for both spatial and temporal solitons. The latter has been well described for optical fibres [4]. For spatial solitons in a planar waveguide, however, the outcome must be a (1 + 1) stable propagation and this is readily modelled by ‘‘freezing” the y-variation, as discussed above. This eliminates the need to specifically introduce a y-dependent modal field variation and assumes that the planar guide supporting the spatial soliton is sufficiently thin to make the modal field a constant across the guide. In general, for a planar guide a TE-mode carries the field components (Ex, Hy, Hz) and a TM mode carries the field components (Hx, Ey, Ez). Furthermore, within the approximation to a thin guide and a ‘‘frozen” y-dependence, the linearly polarised field ^ and H ¼ Hy y ^ . In the, apparently, coupled components are E ¼ Ex x equations, Ex and Hy have the fast variation factored out of them and they are now slowly-varying quantities. However, they do have embedded within them, quantities representing the modal fields that are now the constants, AE and AH, i.e. Ex = B(x, z)AE, Hy = B(x, z)AH. Hence, Hy/Ex is simply AE/AH. Furthermore, for a ‘‘frozen” y-dependence, this ratio is the constant value that comes from a plane wave analysis, in which both x and y variations are absent. For a plane wave with a z-dependence eikz0 z , Maxwell’s equations pffiffiffiffiffiffi show that, using kz0 ¼ xc0 el where c is the velocity of light in a vacuum,

e0 jej l0 jlj

ð2:15Þ

This is the relationship that uncouples Eqs. (2.13) and (2.14). Even if the modal fields are explicitly included, then the integration procedure discussed much earlier can be deployed. This will only result in a modification to the effective width of the guide, or the effective nonlinear coefficients, and will still leave the reducibility of the ‘‘coupled” equations in place, governed by the ratio AE/AH. The uncoupling also applies to temporal solitons in which dispersive effects, as opposed to diffraction, are present. No coupling exists, therefore, and the implied route [34] does not exist. This point has been also recently acknowledged in a different discussion about slow temporal solitons [25], but without making contact with the original [34] assertion of irreducibility. To the correct order, the basic equations for a metamaterial supporting the slowly-varying electric and magnetic field components under discussion are,

 2  2 Hy  e0 jej   ¼ AH ¼ E  AE l0 jlj x   @Ex @ 2 Ex x20 ð3Þ ð3Þ e0 jej þ 2 þ 2 leNL þ elNL 2ikz0 jE j 2 E ¼ 0 @z @x c l0 jlj x x    2 @Hy @ 2 Hy x20 ð3Þ l0 jlj ð3Þ  2ikz0 þ l e e l þ þ Hy  Hy ¼ 0 NL NL @z @x2 c2 e0 jej

ð2:16Þ ð2:17Þ ð2:18Þ

They can be readily generalized to describe saturable and other types of nonlinearity by introducing generic functions f1(jEx j2) and f2(jHyj2), where inclusion of these effects can often arise with special types of dielectric and magnetic responses. In their un-normalised form, Eqs. (2.17) and (2.18) show that the nonlinear coefficients depend upon the choice of operating frequency and could change quite rapidly if a monochromatic beam is launched close to a resonance in the permittivity, or permeability. Eqs. (2.17) and (2.18) can now be generalised by adding in higher-order linear diffraction, nonlinear diffraction and non-paraxiality. It has been shown extensively in previous publications [16,17], that nonlinear diffraction dominates over non-paraxiality, longitudinal field component (vector) effects and also over quintic nonlinearity, if the quintic coefficient is not sufficient to produce the level of saturation needed to prevent beam collapse. In the examples to be given below, therefore, the v(5) contributions to the electric or magnetic nonlinearity will be taken as negligible, and it will be assumed that the nonlinear diffraction is dominant [3,16,17]. Higher-order linear diffraction is retained at this stage because diffraction-management is to be discussed and it remains to be seen whether a managed reduction of the main linear diffraction, for example, will bring the higher-order diffraction into prominence, or not. The magnetooptic influence that will be used later on to control beam behaviour will exploit the Voigt effect and, as will be shown, it requires the use of an asymmetric waveguide structure [36] only, and TM modes. The specific nature of the term to be added to the nonlinear Schrodinger equation to account for the Voigt effect will be addressed in the next section but, at this point, the next set of equations recognises that the nonlinear diffraction of the TE-polarised beam equation depends upon the condition r  E – 0 and that the TM-polarised beam equation depends upon the condition r  H – 0. Hence, before the addition of the term that creates the magnetooptic Voigt influence, the current forms of the nonlinear Schrodinger equations are,

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2.3.1 TE-polarised beams

i

3. Magnetooptic influence

 2

@Ex 1 @ 2 Ex 1 @ 4 Ex 1 x0  þ þ @z 2k @x2 8k3 @x4 2k c2 1 ð3Þ @ 2  2  þ e jEx j Ex ¼ 0 2ek NL @x2

ð3Þ leð3Þ NL þ elNL



e0 jej jE j2 E l0 jlj x x ð2:19Þ

where kz0  k as a notational simplification, and it can be seen that only the dielectric nonlinear polarisation can produce the nonlinear diffraction. Note, that for a double-negative metamaterial l =  jlj and e =  je j and backward waves would exist for which k =  jk j and bright spatial solitons would only be possible for N TE ¼ ðjljeNL þ jejlNL le0 jjeljjÞ < 0. It is interesting, now, that a suffi0 ð3Þ ð3Þ cient condition to make N < 0 is to make eNL < 0 and lNL < 0 but this is not a necessary condition and it is possible to have either ð3Þ

ð3Þ

ð3Þ eð3Þ NL > 0 or lNL > 0 depending on their magnitude, but not simulta-

neously. As a consequence, the coefficient in front of the nonlinear diffraction term can, in principle, be positive or negative. 2.3.2. TM-polarised beams

i

@Hx 1 @ 2 Hx 1 @ 4 Hx 1 x20  3 þ þ 2 4 2k @x 2k c2 @z 8k @x 2   1 ð3Þ @ þ l jH x j2 H x ¼ 0 2lk NL @x2



ð3Þ elð3Þ NL þ leNL



ð2:20Þ

ð3Þ

ð3Þ

ments concerning the signs of k, eNL and lNL also apply to this equation, but in this case N TM ¼ ðjejl

þ jlje

ð3Þ NL

l0 jlj e0 jej Þ

< 0.

Eqs. (2.19) and (2.20) model the propagation of bright spatial solitons in a double-negative metamaterial. For each polarisation, the fast spatial variation is a backward wave, with wave number k =  j kj. The propagation is along the z-axis and this can be scaled by the transformations z = jkjw2Z and x = wX. If no diffraction-management is anticipated at this stage, through which the first-order linear diffraction can be reduced, the higher-order diffraction term can be neglected. It will be reintroduced in Section 4. Both nonlinear Schrödinger equations assume the same generic form

i

 @w 1 @ 2 w @2   jwj2 w  j 2 jwj2 w ¼ 0  2 @Z 2 @X @X

ð2:21Þ

This is the complex conjugate of the standard modified nonlinear Schrödinger equation. It has also been assumed that the nonlinear ð3Þ ð3Þ ð3Þ coefficient in the TE-polarised case is eNL ¼ jeNL j and that lNL is negligible, whilst in the TM-polarised case, it has been assumed that ð3Þ ð3Þ ð3Þ the nonlinear coefficient is lNL ¼ jlNL j, with eNL set as negligible. The reasons for these choices are because for TE beams the nonlinð3Þ ear diffraction is controlled by eNL and for TM beams, it is controlled ð3Þ by lNL . For both polarisations, j ¼ k21w2 , which can be modelled for a particular metamaterial that can be varied by changing the operating frequency of the beams. w has been normalised to the following;

TE : w ¼

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! w2 x20  ð3Þ  jljeNL  Ex 2c2

TM : w ¼

0

l0 jlj jH j2 H e0 jej x x

in which it can be seen that only the magnetic polarisation can produce the nonlinear diffraction and where the vector H = (Hx, 0, 0) has been adopted to characterise the TM beam. The same arguð3Þ NL

Eqs. (2.19) and (2.20) have a form that will be used, later on, for specific applications that bring out the influence of various metamaterial structures upon some useful spatial soliton dynamics. Given some recent literature [37–39], it is clear that metamaterial research is reaching out, very confidently, to the visible optical range so, in this paper, the combination of a homogeneous negative phase metamaterial with a magnetooptical material [40–42] will also be assessed for future applications. This can be achieved using the magnetooptic Voigt configuration, as opposed to the Faraday effect, through the use of the kind of asymmetric waveguide structure shown in Fig. 2 [36,43,44]. For a symmetric structure, the Voigt configuration has no influence upon the guided waves and, even for the asymmetric guide, it is only the TM modes that are affected by the applied magnetic field [36]. Spatial solitons must be TMpolarised, therefore, to engage in any magnetooptic control and for narrow beams it will be the nonlinear magnetic polarisation that determines the nonlinear diffraction. The perturbation v(x)Hx to the nonlinear Schrödinger equation can be derived by using the dielectric substrate magnetooptic tensor, which has the form [43,45]

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! w2 x20  ð3Þ  jejlNL  Hx 2c2

em

n2m

B B ¼B 0 @ 0



0



 n2m

2 iQnm

 

1 0 e 0 C 2 B iQnm C e C ¼ @0

2 A 0  eyz n 0

0

1

eyz C A e

m

ð3:1Þ

where nm designates the refractive index of the substrate, and typically Q could be as small as O(104). Note, also, that all the diagonal elements are set equal to each other. This is not exactly true, but it is a common practical assumption [36] based upon the actual values of the physical parameters. Actually, in the bulk, the Voigt effect is of O(Q2) and is negligible, but it is an interesting possibility that a deliberately created asymmetric guide permits access to an O(Q) Voigt effect provided that TM-polarised spatial solitons are used. In such cases, the metamaterial properties will appear quite naturally through the nonlinear diffraction. For polarised beams, using the tensor shown in Eq. (3.1), the magnetooptic perturbations to the Schrödinger equation are

@Ex ¼ 0; @z @Hx x ¼  n2m QHx TM  polarised : i @z c TE  polarised : i

ð3:2Þ

where Q is an average along the y-axis over the waveguide structure. By definition of the integration along y, this would be zero if

+∞ y

Propagation

z

Air

−∞

Ey

Nonlinear

x

Hx

Magnetooptic

+∞

B0

ð2:22Þ Eq. (2.21) will be used below to discuss the behaviour of a narrow beam in a metamaterial, and it will also be adapted to a discussion of a special kind of diffraction-management.

−∞ Fig. 2. Application of magnetooptic influence through the Voigt effect.

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the structure in Fig. 2. was symmetric. This makes the point, rather beautifully, about the deployment of the Voigt effect to O(Q). The right-hand side of the TM-polarised part of (3.2) must be added to the general equation, (2.20) in order to account for any magnetooptic control. Using the coordinate transformations given above that put the nonlinear Schrödinger equation into dimensionless form means that the appropriate dimensionless form of Eq. (3.2) for a TM-polarised beam is

i

@w x0 ¼ jkjw2 n2m Q w ¼ v w @Z c

first-order linear diffraction is to be reduced by the kind of management discussed below then, in principle, higher-order diffraction could take over, so this possibility also needs to be identified. As shown in Fig. 3, the unit cell is split into two lengths, l1L and l2L, each containing a PPM and an NPM material, respectively. If, for example, the beams is a TE-polarised spatial soliton, then the appropriate form of the nonlinear Schrödinger equations in each part of the unit cell is PPM:

ð3:3Þ

An important point, however, is that if Q is a constant, it will do nothing but add an additional phase shift to the solitary wave solution of (2.21). This can be appreciated by redefining w to be weivz and then noticing that the derivative with respect to z immediately eliminates the magnetooptic term. It is necessary, therefore, to make the magnetooptic parameter some function of X. For example   v ðX Þ ¼ v max sech XX0 , where it is anticipated that the magnetooptic parameter defined in Eq. (3.3) is a function of X that has a maximum value vmax, and that its spread over the X-axis is controlled by a dimensionless half-width, X0. The maximum value, vmax, is associated with the saturation of the magnetization [46]. In fact, it is necessary to adopt a spatial form that causes the magnetization to decline away [47], in both directions, from X = 0. This magnetooptic edition to the nonlinear Schrödinger equation will prove to be very interesting when the numerical simulations are performed. If the spatial soliton is imagined to be like a particle in a well [46] then the X-dependent magnetooptic term is in a position to influence the bright spatial soliton by creating deeper wells or spatial barriers depending upon the direction the applied magnetic field assumes along the X-axis. A full scale, non-reciprocal effect should be expected with one sign of the applied magnetic field permitting bright spatial soliton formation, whilst the other sign can be expected to destroy any soliton creation. 4. Diffraction-management It is a widely accepted procedure to try and manage the dispersion or the diffraction of electromagnetic waves in waveguide structures. Indeed, dispersion-management is a very important technique in optical fibre communications systems [13], and diffraction-management is an important feature of waveguide arrays [15]. It has been suggested that a form of diffraction-management can be arranged using negative phase metamaterials. The latter are used to compensate any phase accumulation in normal positive phase media. This form of diffraction-management has been investigated for ring-cavities for which it has been deployed in a manner that encourages impedance-matching in order to avoid unwanted reflections [18,48,49]. It is possible to imagine a number of impedance-matching scenarios that could involve graded-index materials [50–52], the creation of inhomogeneous nano-structured media, or some other kind of anti-reflection additions to the waveguide structure proposed below. It is therefore assumed for the calculations reported below, that the system is impedance-matched so that any degradation due to inherent mismatch is not considered. This model is adopted here in the form shown in Fig. 3. A nonlinear positive phase medium (PPM), is deposited onto a nonlinear negative phase medium (NPM) in such a way that a soliton beam propagating along the z-axis encounters impedance-free propagation, during which the phase accumulated in the PPM is compensated, completely or, partially, by a negative phase accumulation in the NPM. Given this type of propagation, the unit cell shown in Fig. 4. can be used to develop an averaging process that will then lead to the final form of the slowly-varying envelope equation to be used for the spatial soliton applications discussed below. If the

2i

@Ex 1 @ 2 Ex x2 ð3Þ 1 @ 4 Ex þ 2 eNL1 jEx j2 Ex  3 þ 2 4 @z k1 @x c k1 4k1 @x eð3Þ @ 2  2  þ NL1 jE j E ¼ 0 e1 k1 @x2 x x

ð4:1Þ

NPM:

2i

@Ex 1 @ 2 Ex x2 1 @ 4 Ex  2 0 jNTE jjEx j2 Ex þ    2 4 3 @z jk2 j @x c jk2 j 4k  @x 2

e @  2  þ jEx j Ex ¼ 0 je2 jjk2 j @x2 ð3Þ NL2

2

ð4:2Þ

  ð3Þ ð3Þ where N TE ¼ jljeNL þ jejlNL le0 jjeljj and the subscripts 1 and 2 are 0

used to label the regions of the unit cell. For a TM-polarised beam NTE is simply substituted by NTM. For this polarisation, the final term in (4.1) is absent because in a PPM there is no nonlinear magnetic polarisation. In the NPM, however, the TM-polarisation does permit    ð3Þ

a nonlinear diffraction term, for which the coefficient is l = l jk2 j . NL2

2

Note that both nonlinear diffraction and higher-order diffraction are included for the time being. Building upon the previous discussion, however, it has been assumed, now, that the nonlinear diffraction actually dominates over the other possible contributions from non-paraxiality and the role of quintic nonlinearity in any nonlinear saturation that prevents beam collapse. The average over the unit cell can be effected by considering the types of terms in the Schrödinger equations one at a time and adjusting the coefficients according to whether they refer to the PPM or the NPM part of the cell. In other words, the averages are going to be integrations with respect to z over L and it must be assumed that the unit cell is smaller in scale than a diffraction length measured as k1 w2 or jk2jw2, where w is the width of the spatial soliton. For TE-polarised beams, the outcomes for the averaging are;

+∞ L y

PPM NPM l1L

z x

l2 L

−∞

Fig. 3. A planar waveguide structure consisting of alternating layers of positive phase media (PPM) and negative phase media (NPM). z is the propagation direction and diffraction takes place along the x-axis. Note that the periodic structure has a unit cell of length L.

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0.02

pffiffiffiffiffiffiffiffiffiffi e1 l D ¼ l1  pffiffiffiffiffiffiffiffiffiffi1  l2 e2 l

0.18

2

0.135

0.015

m=3

m=1 0.090

0.005

0.045

κ

0.01

0 0.3

0.4

0.5

0

ω/ωpe Fig. 4. xpm/xpe = 0.6, Drude models assumed for both permittivity and permeabil1  1

1  ity. j ¼ 4p2 m2 1  X12 1  0:6 , where X ¼ xxpe . The red line is the PPM X2 dependence, and the blue curve is the NPM dependence. Left-hand scale, m = 3, right-hand scale, m = 1.

Propagation

1 L

Z

L

0

@Ex @Ex dz  @z @z

ð4:3Þ

Diffraction

1 L

Z

L

0

  2 1 @ 2 Ex 1 k1 @ Ex dz  l1  l2 2 k @x k1 @x2 jk2 j

ð4:4Þ

Nonlinearity

! Z !  2  Z L ð3Þ w20 eNL1 1 l1 L w 1 dz  2 jNTE2 jdz ðjEx j2 Ex Þ L 0 c2 k1 c jk2 j L l1 L   w2 k1 ð3Þ  2 o l1 eNL1  l2 jNTE2 j ðjEx j2 Ex Þ c k1 jk2 j

ð4:5Þ

Nonlinear diffraction

1 L

Z

L

0

!

3Þ eð3Þ k1 eðNL2 eðNL3Þ @ 2  2  1 @2  2  l1 NL1 þ l2 jEx j Ex  jE j E 2 k1 ek @x e1 jk2 j je2 j @x2 x x

ð4:6Þ Higher-order diffraction

1 L

Z 0

0 L

4

1 @ Ex 1 B  3 @l1  3 4k @x4 4k1

l2

ð4:10Þ

2

where ei, li are the respective relative permittivities and permeabilities of the parts of the unit cell shown in Fig. 3a. The strategy for using this equation is to reduce the value of D, so that the first-order diffraction can be minimised, thus permitting an influence from the nonlinear diffraction to appear. If the first-order diffraction is reduced in this way, however, it is possible in principle that the higher-order diffraction term, controlled by the parameter F, is not minimised. Nevertheless, k1/jk2j does not have to be unity, and higher-order linear diffraction can be changed by adjusting the structure. In fact, the kind of compensation to be discussed here, can be targeted at D or F. Indeed, for any choice of D it is possible to arrange for the ratio k1/jk2j to make the contribution of higher-order linear diffraction negligible. This is a fascinating possibility that means that the reduction of D does not necessarily mean that higher-order diffraction has to be introduced. Because of this conclusion, it will not be regarded as a critical contribution to the envelope equation and the emphasis will be directed towards first-order diffraction-management. In addition, it should be noted that the non-paraxiality is also reduced as D tends to zero, but in any case, it is emphasised, once again, that nonlinear diffraction is going to be the dominant influence as beams become narrower and, as shown earlier, takes over the role of preventing beam collapse. Clearly if a nonlinear material is selected that has a large quintic contribution then the latter will compete with the nonlinear diffraction. The latter will still present a limit to beam narrowing, even if the small z-component of the electric field is restored to the argument, it only has a small impact on the role of the nonlinear diffraction. In summary, there are two broad scenarios. In one, there is no diffraction-management but the nonlinear diffraction becomes a very important influence as the beams narrow. The second scenario involves manipulating D, to engage in some form of linear diffraction-management that can be arranged in a number of ways to make D ? D(z) through the agency of a metamaterial. As stated earlier, the scales z = (kw2)Z and x = (w)X can be used, where w is actually an arbitrary unit but can be physically interpreted as the beam width. Hence,

i

 @w D @ 2 w @2  þ jwj2 w þ jTE;TM 2 jwj2 w ¼ 0 þ 2 @Z 2 @X @X

ð4:11Þ

where

1

3 k C @ 4 Ex l2  1 A 4 3 k2  @x

!32

el

1 1  F ¼ l1   e2 l 

ð4:7Þ

Hence, taking the spatial average over the unit cell leads to the master equations

TE:

qffiffiffiffiffiffiffi w ¼ w GTE 1 Ex ; GTE 2 ¼

l1

GTE 1 ¼

  k1 ð3Þ l e  N l ; j j 1 2 TE2 NL1 c2 jk2 j !

x20

3Þ ð3Þ eðNL1 k1 eNL2 þ l2 e1 jk2 j je2 j

ð4:12Þ

TE:

  @Ex @ 2 E x x2 k1 ð3Þ þ D 2 þ 20 l1 eNL1  l2 jNTE2 j jEx j2 Ex @z @x c jk 2 j ! ð3Þ 3Þ 4 eNL1 k1 eðNL2 F @ Ex @2  2   2 þ l þ l jE j E ¼ 0 1 2 4 e1 jk2 j je2 j @x2 x x 4k1 @x

TM:

qffiffiffiffiffiffiffiffi w ¼ w GTM 1 Hx ;

2ik1

ð4:8Þ

TM:

  @Hx @ Hx x k1 ð3Þ l0 jlj 2ik1 þ l e N  l þD j j jH x j2 H x 1 NL1 @z @x2 e0 jej jk2 j 2 TM2 ! ð3Þ  F @ 4 Hx k1 lNL2 @ 2  2    2 þ H H l ¼0 j j 2 x x 4 jk2 j l2  @x2 4k1 @x 2

in which

  x20 k1 ð3Þ l0 jlj GTM ¼ l e j N j ;  l 1 2 TM2 1 NL1 c2 e0 jej jk2 j !

ð3Þ

GTM 2 ¼

k1 lNL2 l2   jk2 j l2 

and for each polarisation

2 0 2 c

ð4:13Þ TE

1

ð4:9Þ

TM

jTE ¼ wG2 2GTE and jTM ¼ wG2 2GTM . 1

Eqs. (4.11)–(4.13) show that there are a number of possibilities for choosing the kind of metamaterial that will support typical bright spatial solitons. The simulations below concentrate, however, on two broadly realistic cases. For TE beams, it will be assumed that the PPM is contributing the major part of the ð3Þ nonlinearity, to such an extent, that in practice, eNL2 ¼ 0 and

A.D. Boardman et al. / Optics Communications 283 (2010) 1585–1597 ð3Þ lð3Þ NL2 ¼ 0 and, of course, lNL1 ¼ 0. Without nonlinear diffraction, it is

not necessary for TM beams to use only magnetic nonlinearity, however, if jTM is required, then it will be assumed that this applies to the case when the nonlinearity is located within the ð3Þ ð3Þ ð3Þ NPM, and that eNL2 ¼ 0, eNL1 ¼ 0 and lNL2 –0. For the diffractionmanaged cases to be presented below, therefore, jTE and jTM reduce to the simple forms

jTE ¼

1 2

k1 w2

;

jTM ¼

1

ð4:14Þ

2

k2 w2

5. Numerical simulation: soliton formation and interaction It is emphasized in this paper, that nonlinear diffraction will prevent beam collapse as the power carried by a spatial soliton increases and results in a limit on how narrow the beams can be. It is a process that dominates non-paraxiality and even quintic nonlinearity [3,16,17], provided the coefficient of the latter is not too large. Nonlinear diffraction is still, nevertheless, a diffraction process so that as beams become narrow, and the power goes up, it will still have to compete with the linear diffraction, unless the latter is managed to be a small influence. These features will be shown in the simulations given below and a special form of diffraction-management will be used. The soliton literature for metamaterials contains examples involving classic cases of modulation instability [21,22,31], self-steepening [20,23,53], self-induced transparency [54], and dark solitons [55]. The question of soliton control has been addressed [56] as has the properties of gap solitons [57]. The latter, however deals with quadratic nonlinear material that does not feature in this discussion. All of the issues addressed here, however, involve only bright spatial solitons driven by third-order nonlinearity. Narrow beam formation will be discussed both in isolation, and also in connection with diffraction-management.

1593

First of all, consider a homogeneous metamaterial planar guide, supporting TE- or TM-polarised bright soliton beams. In this case, jTE ¼ jTM ¼ k21w2 ¼ D = 100%, where D = D  100% and c2

x20 w2 ðeðx0 Þlðx0 ÞÞ

 j, where c is the velocity of light in vacuo. j is

shown in Fig. 4, for a metamaterial model, in which e(x0) and l (x0) have a Drude [58] form. For a typical ratio of xpm/xpe, where xpm and xpe are the effective plasma frequencies, it is seen that j varies rapidly with frequency, and rises and falls with respect to the dispersion-free PPM line. If the beam width is measured as an integral number of wavelengths, i.e. by setting w = mk, then Fig. 4. shows, in contrast to the PPM line, that the variation of the NPM nonlinear diffraction coefficient rises rapidly as the resonance frequency is approached. Note that here the beam width is a function of the wavelength and, therefore, a function of frequency to ensure that a narrow beam is being considered. The information displayed in Fig. 4 can be used to show how a metamaterial of this kind can exercise control over the type of beam that can be generated. Even though the full linear diffraction is used, it is clear that values of nonlinear diffraction coefficient can be found that will be able to compete with the focussing and alter the effective beam shape. It is clear from Fig. 4 that quite different values of nonlinear diffraction coefficient enter into the focussing process when different beam widths are used. The right-hand scale is for a beam width the order of a wavelength, while the left-hand scale is for more typical beam widths and it is this that will be adopted in the later simulations. But first, Fig. 5a shows a first-order bright spatial soliton beam without the influence of nonlinear diffraction. Since the equation has been made dimensionless, the scales are arbitrary. It is then re-simulated using a beam width of the order of a wavelength, and j = 0.17. The result is shown in Fig. 5b and illustrates how to bring nonlinear diffraction into play for an environment in which there is no diffraction-management. It can be seen that a limitation is imposed upon the beam, by lowering its intensity and creating a degree of broadening. Fig. 5c

(a)

(b)

1

0.5

0 -5

0

X

5

(c) Fig. 5. Initial inputs: sech(X), D = 100%, m = 1 (a) j = 0, (b) j = 0.17, (c) Beam profiles after 10 Rayleigh length propagation distance: solid blue line (j = 0), dot–dashed green line (j = 0.1), dashed red line (j = 0.17).

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shows how nonlinear diffraction can influence a first-order spatial soliton, but there is an implication from Fig. 4 that choosing a wider beam would make it difficult to see any changes brought about by nonlinear diffraction on a first-order soliton. This does not mean that such induced diffraction will not act as a serious perturbation to a higher-order soliton. A final comment on this section is that it is not a necessary condition to reduce the beam width dramatically, to access higher values of j, because larger coefficients, nearer to the j resonance region can be obtained with frequency tuning because of the dispersive metamaterial properties. In the absence of diffraction-management, Fig. 6 shows what happens to a higher-order, breathing soliton if nonlinear diffraction is introduced. As stated earlier, for D = 100% it must be expected that a first-order soliton, namely a sech(X) input would need to be a beam with a width the order of a wavelength for the nonlinear diffraction to disturb its stability. Fig. 6a shows, however, that a higher-order soliton, such as 3sech(X) is, in fact, three first-order solitons held together with zero binding energy. Without a perturbation this soliton will propagate down the z-axis as a breather. Fig. 6b shows that it is rapidly perturbed by the presence of nonlinear diffraction. The breathing is interrupted and a narrow, first-order soliton is created, plus some low-energy radiated beams. Once again it should be emphasized that this behaviour could be frequency controlled. In fact, the position on the Z-axis at which the beam starts to split can be controlled for a metamaterial through appropriate frequency tuning. Diffraction-management is created, here, through the use of a double-negative metamaterial and its deployment is sketched in Fig. 3. To begin with, Fig. 7 shows what happens when the diffrac-

tion is reduced from its starting value, on the input plane, to 10% on the output plane. As an example, this reduction is forced to take place in a linear fashion. For comparison, it should be noted that the intensity scale is much more compressed in Fig. 7 than the scale in Fig. 5 j = 0, is an interesting case because it can arise simply because the linear diffraction is dominant. Of course, it is not necessary for j to be exactly zero for the role of nonlinear diffraction to be negligible. Indeed, if frequency-window control is anticipated for a metamaterial, it could arise for values of j at the low end of this window. In contrast, j = 0.0028 is closer to the j resonance region. In both cases, ultra-narrow beam formation occurs, but finite j lowers the peak value because of the added nonlinear diffraction. As shown in Fig. 8 setting the diffraction-management, in order to reduce the linear diffraction to 10%, creates an interesting situation. If a first-order soliton is launched at Z = 0, then the act of decreasing the linear diffraction to 10% means that there is now too much power available for the propagation of a first-order (N = 1) soliton. By analogy [4] with optical fibre work, this act of diffraction reduction should lead to a breathing soliton of order pffiffiffiffiffiffi 10. Indeed, Fig. 8a shows that this is exactly what happens, and a third-order breathing soliton is created. In Fig. 8b, the nonlinear diffraction is introduced into the system, and the result is, as in the higher-order soliton case shown in Fig. 6, that the nonlinear diffraction will act as a perturbation to split the breather. The outcome is three low-power, pseudo-solitons, where the power contained within each is not enough for it to retain its soliton status if it is entered into a normal, 100% diffraction medium, and it will simply radiate. In Fig. 8c, a higher value of the nonlinear dif-

Fig. 6. Initial input: 3sech(X), m = 3, D = 100%. (a) j = 0 and (b) j = 0.0028.

Fig. 7. Initial inputs: sech(X), m = 3. Linear reduction of diffraction-management from an input value of 100–10% over a distance of 10 Rayleigh lengths with (a)j = 0, and (b) j = 0.0028.

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A.D. Boardman et al. / Optics Communications 283 (2010) 1585–1597

Fig. 8. D = 10%. Inputs: w = sech(X). (a) j = 0, (b) j = 0.00168, and (c) j = 0.005.

fraction coefficient has been used and, therefore, as a perturbation, it is larger. This means that the breather is split, now, after approximately 15 Rayleigh propagation lengths, rather than 45. Before a more detailed discussion of magnetooptic control is presented, it is useful to consider, briefly, the basic role played by v(X) in the nonlinear Schrödinger equation. This is demonstrated in Fig. 9 that shows a straight-forward, first-order, bright spatial soliton, in the presence or absence of an applied magnetic field. The form of v(X) is given in the caption and, as pointed out earlier, vmax is proportional to Q, which is the magnetooptic saturation parameter. Typically, Q is the order of 104–103, and this means that for available magnetooptic materials, vmax would be in the range 0.6–6. Hence, an intermediate value vmax = ± 1.8 is used, where the ± indicates the direction of the applied magnetic field along ± X. Briefly, switching the field direction produces an impressive, non-reciprocal, behaviour. The physical explanation of this is that the applied magnetic field raises, or lowers, the effective potential well in which the bright spatial soliton is located. The same effects would appear if a narrow beam was under consideration, as in Fig. 5b when nonlinear diffraction is present. As has been discussed earlier, the effect of applying an external magnetic field to a spatial soliton is to effectively change the depth

Fig. 9. D = 100%, j = 0,

v ðX Þ ¼ v max sech

  X X0

of the potential well in which the soliton appears to find itself. This can be verified for a diffraction-managed metamaterial structure, in which D = 10%. The simulations are shown in Fig. 10. It is emphasized that in order to achieve the well-known Voigt effect, the magnetic field must be directed along the x-direction, and the waveguide must be asymmetric. In Fig. 10a, the input has become a third-order breathing soliton, but it is soon perturbed by the nonlinear diffraction. Upon the application of an applied magnetic field, Fig. 10b shows that this action deepens the ‘‘well” in which the spatial soliton finds itself. As a result, the third-order breathing soliton is fully captured. Upon reversing the applied magnetic field, Fig. 10c shows that the role of the magnetic field is to lift up the floor of the well with the consequence that, not only can a breather not be formed, but a single narrow beam is created with excess energy being rapidly ejected to the left and the right. All of this shows that the magnetic field is a very important control mechanism, and that the combination of magnetooptics, with special nano-structured materials will produce interesting applications. The narrow beam in Fig. 11 that is propagating straight down the z-axis could be useful to capture for data processing and other integrated optics applications [59]. Unfortunately, the beam pro-

. Input: w = sech(X), (a)

vmax = 0, (b) vmax =

+ 1.8, and (c)

vmax =

 1.8.

  Fig. 10. D = 10%, j = 0.00168, m = 3, v ðX Þ ¼ v max sech XX0 . Input: w = sech(X), (a) v(X) = 0, (b) v(X) > 0, and (c) v(X) < 0.

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y

Beam

y

PPM

Beam

NPM

z

z

x

B0

x Magnetooptic

(a)

Fig. 11. w = sech(X), j = 0.00168, D = 10% for 0 < Z < 100, D = 100% for Z > 100. (a) Sketch of a partial diffraction-managed waveguide with a magnetooptic end-section, (b) vmax = 0, and (c) vmax = + 1.8.

duced from the diffraction-management does not contain enough power to sustain its shape when it is projected into a 100% diffraction medium. This is exactly displayed in Fig. 11b, where, as shown, the first 100 Rayleigh lengths is a 10% diffraction-managed medium, and beyond that distance, the linear diffraction is increased to 100%. This causes the beam emerging from the diffraction-managed region to diffract away and become lost. In Fig. 11c, this emerging beam is fired into a magnetooptic region, so that an external magnetic field is applied. The result is that the beam is now captured in a deep well, and could be used in an application for further processing. It is assumed that the magnetooptic region is the asymmetric waveguide configuration considered earlier on, and that the kind of impedance-matching in place throughout the diffraction-managed region permits the beam to enter freely into the magnetooptic region. 6. Conclusions A major study of TE and TM linearly polarised, bright, spatial solitons in double-negative homogeneous metamaterials has been presented, that embraces both new and existing work in the field. The outcomes anticipate the use of planar metamaterial waveguides of the future, which will be nano-structured and be capable of operating in the range of visible to near infra-red frequencies, where magnetooptic effects can be exploited. In the light of recent work addressing losses in metamaterials [26–29], loss has not been added into the nonlinear Schrödinger equation, but, as has been pointed out recently [25], practical soliton applications, using metamaterials, will not be impossible to achieve. A recognition is given of the growing interest in soliton phenomena in metamaterials and the basic methodology for studying polarised soliton beams is presented in some detail and nonlinear diffraction is discussed extensively. The ideas are applied to homogeneous planar

waveguides containing (1 + 1) spatial solitons, and these are then extended to investigate a form of diffraction-management. To all of this is added magnetooptic control that, interestingly, only uses the Voigt, or Cotton–Mouton, effect. It is shown how the latter depends upon an asymmetric guiding structure, and it is clear from the outcomes that an important degree of magnetic control can be achieved. All of the theoretical development is supported by strong numerical simulations and it is clear that this type of metamaterial environment is going to be very valuable for applications.

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