Coupled nonlinear Schroedinger field equations for electromagnetic wave propagation in nonlinear left-handed materials

1.2 MB·English

by  N. Lazarides

Coupled nonlinear Schr¨odinger field equations for electromagnetic wave propagation
in nonlinear left-handed materials
N. Lazarides 1,2 and G. P. Tsironis 3
1Department of Materials Science and Technology, University of Crete, P. O. Box 2208, 71003, Heraklion, Greece
2Department of Natural Sciences, Technological Education Institute of Crete, P. O. Box 1939, 71004, Heraklion, Greece
3Department of Physics, University of Crete, and FORTH, P. O. Box 2208, Heraklion, 71003, Greece
showed that the magnetic field intensity couples to the
For an isotropic and homogeneous nonlinear left-handed
magnetic resonance of the SRR in a nontrivial way, and
materials, for which the effective medium approximation is
thatchangingthe materialproperties fromleft- to right-
5 valid,Maxwell’sequationsforelectricandmagneticfieldslead
handedandbackisallowedbyvaryingthefieldintensity.
0 naturally, within the slowly varying envelopeapproximation,
The study of the nonlinear properties of LHMs could fa-
0 toasystemofcouplednonlinearSchr¨odingerequations. This
cilitatefutureeffortsincreatingtunablestructureswhere
2 system is equivalent to the well-known Manakov model that
n undercertainconditions,iscompletelyintegrableandadmits the field intensity changes their transmission properties.
In the present work we show that for an isotropic,
a brightanddarksolitonsolutions. Itisdemonstratedthatleft-
homogeneous, quasi-one-dimensional LHM, Maxwell’s
J andright-handed(normal)nonlinearmediumsmayhavecom-
equations with nonlinear constitutive relations lead nat-
1 pound bright and dark soliton solutions, respectively. These
urally to a system of coupled nonlinear Schr¨odinger
3 results are also supported bynumerical calculations.
(CNLS) equations for the envelopes of the propagating

] electricandmagnetic fields. Thissystemis equivalentto
S Recently the study of the electromagnetic (EM) prop- the Manakov model10 that under certain conditions, ad-
P erties of artificial complex media with simultaneously
mitssolitonsolutionsconsistingoftwocomponents(vec-
. negative dielectric permittivity ǫ and magnetic per-
n eff torsolitons). Forspecific parameterchoices,correspond-
i meability µeff has been the subject of great attention. ing to either a left-handed or a right-handed medium,
l
n Suchmediaareusuallyreferredtoasleft-handedmateri- we find compound dark and bright soliton solutions, re-
[ als(LHMs),1 andtheydemonstrateanumberofpeculiar
spectively. The constitutive relations can be generally

properties: reversal of Snell’s law of refraction, reversal
1 written as
of the Doppler shift, counter-directed Cherenkov radia-
v
2 tioncone,negativerefractionindex,therefocusingofEM D=ǫeffE=ǫE+PNL (1)
05 wpraovpeesrtfireosmfoallopwoidnitrescotluyrsfer,ometcM.axTwheell’asbeoqvuea-mtioenntsiowniethd B=µeffH=µH+MNL, (2)
1 appropriateconstitutiverelations. Pendry2hasproposed where E and H are the electric and magnetic field in-
0 the intriguing possibility to exploit the negative refrac- tensities, respectively, D is the electric flux density, and
5
0 tion index property of the LHMs in order to overcome B is the magnetic induction. The linear dielectric and
known problems with common lenses to achieve a per-
/ magnetic responses of the LHM are described by ǫ and
n fect lens that would focus both the propagating as well µ, respectively, while P = ǫ E and M = µ H
li as the evanescent spectra. are the nonlinear electriNcLpolarNizLation and tNheLnonlNinLear
n
TypicalLHM are composedof a combinationofa reg-
magnetization of the medium, respectively.
:
v ulararrayofelectricallysmallresonantparticlesreferred Itisknownthatǫ andµ inaLHMhavetobedis-
eff eff
Xi toof caosndspulcitt-inrigngwirreesso,2n–a6torressp(oSnRsRib’lse)foarndthae nreegguatlaivreaµrray persive,otherwisethe energydensitycouldbe negative.1
eff Their frequency dispersion, including nonlinear effects
r
a and ǫeff, respectively. The size and spacing of the con- (but neglecting losses), is given by7
ducting elements of which the medium is composed is
assumed to be on a scale much smaller than the wave- ω2
lengths in the frequency range of interest, so that the ǫ (ω)=ǫ ǫ (E 2) p (3)
composite medium may be considered as a continuous eff 0 D | | − ω2!
and a homogeneous one (effective medium approxima- Fω2
µ (ω)=µ 1 , (4)
tion). Thus far, almost all properties of LHMs were eff 0 − ω2 ω2 (H 2)
studied in the linear regime of wave propagation, when (cid:18) − 0NL | | (cid:19)
both ǫeff and µeff are considered to be independent of where ωp is the plasma frequency, F is the filling factor,
the field intensities. However, nonlinear effects in LHMs ω = ω (H 2) is the nonlinear resonant SRR fre-
0NL 0NL
have been recently taken under consideration by some quency,andǫ |(E|2)=ǫ +αE 2,withαthe strength
authors.7–9 Zharov et al,7 considered a two-dimensional of the nonlineDari|ty.| PositDiv0e (ne|ga|tive) α corresponds to
periodic structure created by arrays of wires and SRR’s a focusing (defocusing) dielectric. For a linear dielectric
embeddedintoanonlineardielectric,andtheycalculated ω (H 2) ω , where ω is the linear resonant SRR
0NL 0 0
| | →
ǫeff andµeff foraKerr-typedielectricpermitivity. They frequency. Then Eqs. (3-4) reduce to previously known
1
expressions.5,11TheparametersF,ω ,andω arerelated ξ =ε(z ω′t) τ =ε2t, (10)
p 0
−
togeometricalandmaterialparametersoftheLHMcom-
′
ponents. Although ǫ can be readily put in the form where ε is a small parameter, and ω = ∂ω/∂k is the
eff
ǫ+ǫ (E 2), for µ this is not an obvious task, since groupvelocityofthe wave. Takinginto accountEqs. (8-
NL eff
µ =µ| |(ω ), and ω depends on H 2 as7,8 9), substituting slow variables into Eqs. (6-7), assuming
eff eff 0NL 0NL
| | that α = αˆε2, β = βˆε2, and expressing p and q as an
αΩ2X6|H|2 =A2Ec2(1−X2)(X2−Ω2)2, (5) asymptotic expansion in terms of ε12
where X = ω /ω , Ω = ω/ω , E is a characteris-
0NL 0 0 c q(ξ,τ)=q (ξ,τ)+εq (ξ,τ)+ε2q (ξ,τ)+
tic (large) electric field, and A is a function of physi- 0 1 2 ···
cal and geometrical parameters.7,8 The + (-) sign cor- p(ξ,τ)=p0(ξ,τ)+εp1(ξ,τ)+ε2p2(ξ,τ)+ , (11)
···
responds to a focusing (defocusing) dielectric. Our α is
we get various equations in increasing powers of ε. The
related to the parameters of Eq. (5) as α = 1/E2.
± c leading order problem gives the dispersion relation ω =
Choosing f = ω /2π = 10 GHz and f = ω /2π =
p p 0 0 ck, where c = 1/ǫµ. At (ε1), the group velocity is
1.45 GHz, left-handed behavior appears in a narrow given as ω′ =kc2/ω. At (εO2), we obtain
frequency band (from f = 1.45 GHz to 1.87 GHz). p O
For relatively small fields when µeff is trully field de- ∂q ω′′∂2q ωc2
pendent, one may consider that the magnetic nonlin- i 0 + 0 + (αˆµq 2+ǫβˆp 2)q =0 (12)
earity µ = µ + µ (H 2) is of the Kerr type, i.e. ∂τ 2 ∂ξ2 2 | 0| | 0| 0
eff NL
µneNtLic(|nHon|2l)in=eaβrit|yHβ|2c(aFnigb.e| t1r|)e.atTehdeasstarefintgttinhgopfatrhaemmetaegr-. i∂∂pτ0 + ω2′′∂∂2ξp20 + ω2c2(ǫβˆ|p0|2+αˆµ|q0|2)p0 =0, (13)
where ω′′ = (c2 ω′2)/ω, τ is the slow time, and ξ the
−
0 slow space variable moving at the linear group velocity.
By rescaling τ, ξ and the amplitudes q , p according to
0 0 0
µµ/0eff−0.5 µµ/0eff ξ =X, T =ω′′τ/2, (14)
−1 −0.70 |H|20 /.0 E5c2 0.1 Q= |Λq/ω′′|q0, P = |Λp/ω′′|p0 (15)
q q
0 0.5 1
|H|2 / E2 where Λ =ωc2µaˆ and Λ =ωc2ǫβˆ, we get
c q p
FIG.1. µ as a function of |H|2/E2, for Ω=1.1 (solid),
eff c
1.15 (dotted), 1.20 (dashed), 1.25 (long-dashed), A = 3, iQT +QXX + σq Q2+σp P 2 Q=0 (16)
| | | |
F = 0.4, and αˆ > 0. Inset: Fitting to a line of the three iPT +PXX + (cid:0)σp P 2+σq Q2 (cid:1)P =0, (17)
first curves of themain figure for relatively small fields. | | | |
whereσ sign(Λ(cid:0) ). Eqs. (16-17(cid:1))isaspecialcaseof
q,p q,p
Using Eqs. (1)-(2) and known identities, we get quite ≡
thefairlygeneralandfrequentlystudiedsystemofCNLS
general vector wave equations for the fields E and H
equations (Manakov model) known to be completely in-
∂2E ∂2P tegrableforσq =σp =σ.10 Anumber ofbrightanddark
2E µǫ µ NL ( E)= soliton solutions have been obtained for Eqs. (16-17)
∇ − ∂t2 − ∂t2 −∇ ∇· when σ = 1.14–19 There is evidence that single-soliton
∂ ∂ ∂ ±
[( µ ) H]+ µ (ǫE+P ) (6) (single-hump) solutions are stable while multi-hump are
NL NL NL
∂t ∇ × ∂t(cid:20) ∂t (cid:21) not.16 Thesignofthe productsµαˆ andǫβˆdetermine the
2H µǫ∂2H ǫ∂2MNL ( H)= type of nonlinear self-modulation (self-focusing or self-
∇ − ∂t2 − ∂t2 −∇ ∇· defocusing) effects which will occur. For σ = 1 both
±
∂ ∂ ∂ fields experience the same type of nonlinearity.
−∂t[(∇ǫNL)×E]+ ∂t ǫNL∂t(µH+MNL) . (7) For ǫ, µ > 0 and aˆ,βˆ > 0 we have σ = +1 , and the
(cid:20) (cid:21) system of Eqs. (16-17) accepts solutions of the form15
Consider an x polarized plane wave with frequency ω
propagating alo−ng the z− axis:12 Q(X,T)=u(X)eiνq2T P(X,T)=v(X)eiνp2T, (18)
E=(E(z,t),0,0), H=(0,H(z,t),0), (8)
where u, v are real functions and ν , u are real posi-
q p
where tive wave parameters. The latter is necessary, if we are
interested in solitary waves that exponentially decay as
E(z,t)=q(z,t)ei(kz−ωt) H(z,t)=p(z,t)ei(kz−ωt), (9) X . IntroducingEq. (18)into Eqs. (16-17)weget
| |→∞
with k being the wavenumber. The envelopes q(z,t) and u ν2u+(u2+v2)u=0 (19)
p(z,t) of E and H, respectively, change slowly in z and XX − q
t. We therefore introduce the slow variables vXX −νp2v+(v2+u2)v =0. (20)
2
1.5 relative amplitudes are controlled by the corresponding
nonlinearities and frequency. For σ = 1 Eqs. (16-17)
p|0 1 have also solutions of the form19 −
|q|, |00.5 Q(X,T)=u(X)e−iνq2T P(X,T)=v(X)e−iνp2T, (26)
0
−8 −4 0 4 8 where u, v are real functions and νq, νp are real positive
X
wave parameters. Introducing Eqs. (26) into Eqs. (16-
FIG.2. Envelopesq0andp0ofthecompoundbrightsoliton 17), we get
(σp = σq = +1) for ν = 1 and r = Λq/Λp = 2 in arbitrary
units, with themaximum amplitude of p0 normalized to 1. u +ν2u (u2+v2)u=0 (27)
XX q −
v +ν2v (v2+u2)v =0. (28)
For νq,p = ν, Eqs. (19-20) have a one-parameter family XX p −
of symmetric and single-humped soliton solutions (Fig.
2)15–17 Forνq,p =ν,Eqs. (27-28)allowforkink-shapedlocalized
soliton solutions20
u(X)=±v(X)=νsech(νX). (21) u(X)= v(X)=(ν/√2)tanh(νX/√2), (29)
±
There are also periodic solutions of the form
ascanbeseeninFig. 4. Inthecontextofthepropagation
oftwo polarizationcomponentsof a transverseEMwave
u(X)=Acos(BX) v(X)=Asin(BX), (22)
in a Kerr-type medium, the solution of this kind is often
calledapolarizationdomainwall. Therearealsoperiodic
where A = √ν2+B2 with B an arbitrary parameter.
solutions of the form
Nowwithoutlossofgenerality,wetakeν =1anddenote
q
ν as ν. This can always be acheived by a rescaling of
p u(X)=Acos(BX) v(X)=Asin(BX), (30)
variables u, v and X. Then, for 0 < ν < 1 there is
another, in generalasymmetric, one-parameterfamily of where A = √ν2 B2 with B < ν an arbitrary parame-
solutions for each fixed ν15,18 ter. −
u(X)= 2(1 ν2)cosh(νX)/κ (23) 1.5
−
v(X)=pν 2(1 ν2)sinh(X X0)/κ, (24)
− − −
where p q|, |p|00 0
|
κ=cosh(X X )cosh(νX) νsinh(X X )sinh(νX)
− 0 − − 0 −1.5−8 −4 0 4 8
X
where X is an arbitrary parameter. For X = 0, u
0 0 FIG.4. Envelopesq0 andp0 ofthekink-shapedcompound
becomes symmetric and v antisymmetric.
soliton (σp = σq = −1) for ν = 1 and r = Λq/Λp = 2 in ar-
bitraryunits,with themaximumamplitudeofp0 normalized
2
to 1.
p|0 Inorderto checkthe validity ofthe Kerr-typeapprox-
q|, |01.5 imation for the magnetic nonlinearity, we performed ex-
|
tended numerical simulations of the complete Eqs. (6 -
1 7)for the fields of Eqs. (8 -9) using the full expression
−8 −4 0 4 8
X for µ , with ω obtained analytically from Eq. (5),
eff 0NL
FIG.3. Envelopesq0 andp0 ofthecompounddarksoliton and slightly different normalization. In the simulations
(σp = σq = −1) for k = 1 and r = Λq/Λp = 2 in arbitrary we used the exact solitons of Eq. (21) for right-handed
units, with theminimum amplitude of p0 normalized to 1. medium (RHM) and Eq. (25) for LHM as initial con-
ditions and tested their subsequent time evolution and
For ǫ, µ < 0 and aˆ,βˆ > 0 we have σ = 1 , and resulting stability. The Q-field amplitudes are shown in
Eqs.(16-17)accept dark soliton solutions of the−form19 Fig. 5 fora right-(two rightfigures)anda left- (two left
figures) handed medium with Ω = 1.2 and 3.0, respec-
Q(X,T)=P(X,T)=k[tanh(kX) i]ei(kX−5k2T), (25) tively. We take the (average)amplitude ofQto be unity
−
forthe right-handedmedium, while the amplitude ofthe
whicharelocalizeddipsonafinite-amplitudebackground background wave of Q is unity for the LHM. Similar re-
wave, as shown in Fig. 3. In this very interesting case sults are obtained for the field P. In the left part of Fig.
of LHM the electric and magnetic fields are coupled to- 5,twobrightsolitonsolutionsoftheformofEq. (21)are
gether forming a dark compound soliton. Note that the shown, with their difference being the chosen value of ν
3
in the initial conditions. For relative high ν the soliton choice of parameters corresponding to LHM, the system
is still stable, but its amplitude is strongly oscillating. admits compound dark soliton solutions. For the choice
Forrelativelylowν the solitonispropagatingpractically ofparameterscorrespondingto normalmedium, the sys-
undisturbed. In the right part of Fig. 5 two dark soli- tem admits compound bright soliton solutions. Refer-
ton solutions of the form of Eq. (25) are shown, with encetocompound”dark”and”bright”solitonswemean
their difference being the chosen value of the constant that both soliton components, i.e. both the envelopes of
k in the initial conditions. Again, for relatively high k the electric and the magnetic fields, areeither both dark
the soliton developes strong osicllations and deforms as or both bright. The described effects are not limited
time progresses while for relatively low k the soliton is to the specific SRR - wire system. They should also be
propagating practically undisturbed. The numerics thus presentinothernonlinearLHMdesigns,suchasphotonic
demonstratesclearlythattheanalyticalsolutionsofEqs. crystals,22 coupled nanowiresystems,23 the transmission
(12-13)are good approximate solutions for the complete line systems,24 and photonic systems.25 The case where
problem at relatively small fields while at larger ampli- thefieldsinthemediumexperiencedifferenttypeofnon-
tudes the exact solitons deform. This is expected since linearity, leading to σ = σ =1 or σ = σ =1 cor-
q p p q
− −
magnetic nonlinearity ceases to be of Kerr-type and sat- respondsto amedium of positiveǫ andnegative µ
eff eff
uration effects become more important. or negative ǫ and positive µ , respectively. This in-
eff eff
teresting case where the Manakov system does not seem
tobeintegrablewillbetreatednumericallyinafollowing
publication.
|Q| |Q|
1
1
0.5 0 1V.G. Veselago, Sov.Phys. Usp. 10, 509 (1968).
5000 5000 2J. B. Pendry,Phys. Rev.Lett. 85, 3966 (2000).
-1000 T -80 T 3D. Smith, W. Padilla, D. Vier, S. Nemat-Nasser, and S.
X 0 0 X 0 0 Schultz,Phys.Rev. Lett.84, 4184 (2000).
1000 80 4R.Shelby,D.Smith,andS.Schultz,Science292,77(2001).
5J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J.
Stewart, IEEE Trans. Microwave Theory Tech. 47, 2075
|Q| |Q| (1999); J. B. Pendry, A. J. Holden, W. J. Stewart, and I.
Youngs, Phys. Rev. Lett. 76, 4773 (1996); J. B. Pendry,
1
1 A. J. Holden, D. J. Robbins, and W. J. Stewart, J. Phys.:
Condens. Matter 10, 4785 (1998).
0.7 0 6R.M.Walser,A.P.Valanju,andP.M.Valanju,Phys.Rev.
Lett. 87, 119701 (2001).
5000 5000
7A.A.Zharov,I.V.Shadrivov,andY.Kivshar,Phys.Rev.
-5000 T -400 T Lett. 91, 037401 (2003).
X 0 0 X 0 0 8S. O’Brien, D. McPeake, S. A. Ramakrishna, and J. B.
5000 400
Pendry,Phys. Rev.B 69, 241101 (2004).
FIG. 5. Space-time evolution of solitons. Right: Bright 9M.Lapine,M.Gorkunov,andK.H.Ringhofer,Phys.Rev.
solitons(RHM)withhigh(upper)andlow(lower)amplitude E 67, 065601 (2003).
ν. Left: Dark solitons (LHM) with high (upper) and low 10S.V. Manakov, Sov.Phys.-JETP 38 248 (1974).
(lower) amplitude k. 11M. Gorkunov, M. Lapine, E. Shamonina, and K. H.
Ringhofer, Eur. Phys. J. B 28, 263 (2002).
In conclusion, we obtained a system of CNLS equa- 12C.V.Hile,WaveMotion 24,1(1996); M.P.Sorensen,M.
tions equivalent to the Manakov model describing the Brio,G.M.Webb,andJ.V.Moloney,PhysicaD170,287
propagationofEMwavesinanonlinearLHM(aswellas (2002).
in a nonlinear RHM) for relatively small fields. Unlike 13D. J. Kaup and B. A. Malomed, Phys. Rev. A 48, 599
a recent article,21 where only magnetic nonlinearity was (1993).
considered in the propagation of EM waves in LHMs, in 14Q-H.Park and H.J. Shin, Phys.Rev.E 61, 3093 (2000).
thepresentworkbothmagneticanddielectricnonlinear- 15J. Yang, Physica D 108, 92 (1997).
ities were retained. The present analysis however does 16M. H.Jakubowski, K.Steiglitz, and R.Squier,Phys.Rev.
notaddresstheissueoftheswitchingeffectbetweennor- E 58, 6752 (1998).
17M. Haelterman and A. Sheppard, Phys. Rev. E 49, 3376
malmedium(RHM) andLHM properties. Althoughnot
(1994).
obviousfrom the Eqs. (3 - 5), the reductionto the Man-
18D. N. Christodoulides and R. I. Joseph, Optics Lett. 13,
akovformbecomespossibleafteraproperapproximation
53 (1988).
ofthecomplexEqs. (4)and(5). Itturnsoutthatforthe
19Y. S. Kivshar and S. K. Turitsyn, Optics Lett. 18, 337
4
(1993); A. P. Sheppard and Y. S. Kivshar, Phys. Rev. E
55, 4773 (1997).
20Y. S. Kivshar and B. Luther-Davies, Phys. Rep. 298, 81
(1998).
21I.V.ShadrivovandY.S.Kivshar,arXiv:physics/0405031.
22P.MarkoˇsandSoukoulis,Phys.Rev.E65,036622 (2002).
23V.A.Podolskiy,A.K.Sarychev,andV.M.Shalaev,Optics
Express 11, 735 (2003).
24G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer, IEEE
Trans. Microwave Theory and Techniques 50 2702 (2002).
25G. Shvets,Phys. Rev.B 67, 035109 (2003).
5