Theoretical Model

Modulation instability in Metamaterials embedded into a non-Kerr medium*

4.2 Theoretical Model

The pulse propagation equation in nonlinear MM with both χ⁽³⁾ and χ⁽⁵⁾ nonlinearities are obtained by generalizing the derivation of the so called Nonlinear Schrodinger equation (NLSE) based on previous works [91-92]. It is worthwhile to mention that one of the most fundamental differences between an ordinary material and an MM is that MM exhibits strong dispersive behavior in electric permittivity and magnetic permeability, while, in ordinary materials, only one of them is dominant at a time. We consider that the MM, embedded in cubic-quintic non-Kerr medium, is characterized by the nonlinear electric polarization (P_NL) as P_NL =ε_NLE=ε χ₀ ⁽³⁾|E|² E+ε χ₀ ⁽⁵⁾|E|⁴ E; where E is the electric filed, ε_NL is the nonlinear electric permittivity and χ^{( )n} is the n-th order electric susceptibility. The dielectric permittivity (ε) and magnetic permeability (µ) are dispersive in MMs and their frequency dispersion is given by Eq.(3.1) as from the so called lossy Drude model [133]. The recent work of Stockman [152] shows that loss is an extremely important issue in MMs and in this work we are taking the electric and magnetic normalized loss parameter as

e m 0.01

γ ^∼γ ^∼ , a value two orders of magnitude greater than that of the one taken by D’Aguanno et al. [133,152].

Now, starting with Maxwell’s equations and adopting a procedure similar to that of Ref. [92], we obtain the following one dimensional pulse propagation equation in a nonlinear negative index metamaterial:

( ) ( )

( )

2 3 4 2

2 2 2

3

2 4

0 1 2

2 3 4 2

4 4

0 0 3

| | | | | |

2 6 24

i i

A A A A

i A A i S A A S A A

i A A S A A

β

β β

ξ τ τ τ σ τ τ

η η

τ

 

∂ ∂ ∂ ∂ ∂ ∂

= − + + +  + − 

∂ ∂ ∂ ∂  ∂ ∂ 

′ ∂

+ −

∂

(4.1)

In Eq. (4.1), A is the slowly varying envelope of the pulse, propagating along the ξ direction.

2, 3

β β and β₄ are the group velocity dispersion(GVD), the third order dispersion(TOD) and the fourth order dispersion(FOD) parameters respectively, while σ₀ and η₀^′ are the cubic and quintic nonlinear coefficients respectively. S₁ is the so-called self-steepening (SS) parameter due to cubic nonlinear polarization while S₃ is the SS parameter due to quintic nonlinear polarization. On the other hand, S₂ is the second order nonlinear dispersive co-efficient. The above mentioned parameters are defined as follows:

2 2

2 2 0 0 3 3 2 0 4 4 2 0 3 0

1 0 1 0 0 2 2 0 1 0 0 0 0 0 1 0 0

3 0 1 0 0

2 1 ; 3 ; 3 4 ;

1 1 ; 4 1 ;

1 1

k k V k V k k V

S i k V S i k k V i k V

S i k V

β δ β δ β β δ β δ

ω σ σ σ σ σ σ ω β ω σ σ

ω η η

= − = − = − −

= − − = − − − − +

= − −

(4.2)

with

0 0

0 0

0 0

! 1 ( ) ( )

; 2 / ( ( ) ( ) );

2 ! !

l m l

m

m l m l

l

m V k

k l m l _{ω ω ω ω}

ωε ωµ

δ ω ε ω γ µ ω α

ω ω

−

−

= = =

∂ ∂

= = +

− ∂ ∂

∑

(3) (5)

0 0 0 0 0 0

! / 2 ; ! / 2

m m Fm k m m E Fm k

σ = ω ε χ η = ω ε χ , (4.3) where

0

( )

!

m m

m m

F i

m _{ω ω}

ωµ

ω ₌

= ∂

∂ ,k₀ =nω₀/c,

{ }

0

( ) / α ωε ω ωω ω

= ∂ ∂ = and

{ }

0

( ) / γ ωµ ω ωω ω

= ∂ ∂ =

The non-SVEA correction terms to first order is approximated as:

2

2 4 2

4 2

2 0 2

2

2 4 0 2

2 3 4 2

2 4 2

3 0

2

0 0

3 4 2

2 0 2

1 2

4 2

2 6

A A

A A A A

A i A A

i A A i A A A A

i A A

β σ β

ξ τ σ τ

δ σ

β σ η

ξ τ τ τ τ τ ω τ

σ τ

∂ ∂ ∂

≈ − − +

∂ ∂ ∂

∂ ∂ ∂ ∂ ∂ ∂

≈ − + + + −

∂ ∂ ∂ ∂ ∂ ∂ ∂

+ ∂

∂

(4.4)

It is clear that S₂term arises, due to the dispersive nature of electric permittivity and magnetic permeability, which is not present in conventional materials. It is worthwhile to mention that the FOD parameter β₄contains one extra term while the second order nonlinear dispersive parameter S₂ has two additional terms as compared to the ones considered in an earlier work [144]. This difference could be attributed to the fact that, in this work, we are considering both the cubic and the quintic nonlinearities. In the expression forβ₄, we are getting an additional term due to the TOD effect. In our model equation, we are getting one additional term associated with S₃which represents the self-steepening effect arising due to quintic nonlinearity. In Fig.4.1, we plot the variation of the refractive index n, β β₂, ₃ and β₄ with the normalized frequency ω ω₀/ _pe for ω_pm/ω_pe =0.8. Similar plots are depicted for σ₀ and η₀^′ in Fig. 4.2(a), and for S₁and S₂ in Fig. 4.2(b). All the parameters are calculated at

Fig.4.1: Plot of the refraction index (n), group velocity dispersion (β₂), the third order dispersion parameter (β₃) and the fourth order dispersion parameter (β₄) versus the normalized frequency

ω ωpewith ω_pm =0.8_.

ω =ω0and we have taken γ_e ^∼γ_m ^∼0.01. It should be noted that β β₂, ₃ and β₄are plotted in the units of 1 cω_pe, 10² cω²_peand 10³ cω³_pe respectively while σ₀ is calculated in units of

(3) /

pe c

ω χ and η₀′ is calculated in units ofω_pe(χ^{(3) 2}) /c. We have chosen

(

^χ(3)

)

^{2 ≈ χ(5) in}

Fig.4.2(a). S₁, S₂andS₃ are calculated in the units of 1ω_pe , 1/ω²_peand 1 /ω_pe respectively.

We have studied the parameters both in the negative and the positive index regime. In the negative index region,0<ω ω_{0 pe} <0.8, β β β₂, ₃, ₄,S₁andS₂can be either positive or negative, but in the positive index region, ω ω_{0 pe} >1, β β₂, ₄,S₁andS₂are always negative and β₃ is always positive. Both σ₀ andη₀′ are positive in the negative and the positive index region. The behavior ofS₃ is similar to that of the self-steepening parameterS₁, as could be seen from Eq.(4.2). However, the only difference is that S₃ arises due to quintic nonlinear polarization, while S₁results due to the cubic nonlinearity.

Fig.4.2: (a) Plot of the refraction index (n),the third order nonlinear polarization parameter(σ₀) and the fifth order nonlinear polarization parameter (η₀′) versus the normalized frequency ω ω_pewith

pm 0.8

ω = b) Plot of refraction index (n),the first order SS parameter(S₁) and second order nonlinear parameter(S₂) versus the normalized frequency ω ω_pewith ω_pm=0.8_.

In our analysis the second order nonlinear dispersion term S₂ contains two additional terms compare with the earlier results [144] which arises due to the proper approximation of the non-SVEA correction terms. As a result of this as the normalized frequency increases S₂ parameter decreases from positive to negative value in the negative index region as shown in Fig.4.2(b).

For the purpose of simplification it is convenient to write Eq. (4.1) in the normalized form.

We assume the normalized variables as : Z =ξ L_D,T =τ T₀, A= P U₀ , u=N U₀ . Here

2

0 / 2

LD =T β is the so-called second order dispersion length, T₀is the pulse width. We can similarly define the third and the fourth order dispersion length as L_D′ =T₀³/ β₃ and

4

0 / 4

LD′ =T β respectively. N₀ is termed as the order of the soliton, defined as N₀² =L_D L_NL with L_NL ⁼1/σ_{0 0}P , the so-called nonlinear length.

Eq. (4.1) can thus be written in the dimensionless units as follows:

( )

( ) ( )

2 3 4

2 4 2

3

2 4

2 3 4 1

2 2 2

2 2 3

| | | |

2 6 2 4

b

i b i b

u u u u

i u u i p u u s u u

Z T T T T

i s u u s p u u

T T

∂ ∂ ∂ ∂ ∂

= − + + + + −

∂ ∂ ∂ ∂ ∂

∂ ∂

− − ′

∂ ∂

(4.5)

Here b₂ =sgn(β₂) ,b₃ =L_D/L′_D and b₄ =L_D/L′′_D . s₁= S₁ /T₀ and s₂ = S₂ T₀² are the normalized SS parameter and second order normalized nonlinear dispersion parameter respectively, while s₃ = S₃ T₀ is the normalized SS parameter due to quintic nonlinearity.

2

0 0 _D 1 2 0 _D

p=η σ′ L − k L is the normalized quintic nonlinear parameter andp′=η σ₀′ ₀²L_D.