1) The document discusses travelling wave solutions for pulse propagation in negative index materials (NIMs) in the presence of an external source. 2) It obtains fractional-type solutions containing trigonometric and hyperbolic functions by using a fractional transform to map the governing equation to an elliptic equation. 3) Specific solutions include dark/bright solitary waves described by a sech-squared profile, as well as periodic solutions.

1. Travelling wave solutions in negative index materials in the
presence of external source
Vivek Kumar Sharma∗ , Amit Goyal∗ , C. N. Kumar∗ and J. Goswamy†
∗
Department of Physics, Panjab University, Chandigarh 160 014, India
†
UIET, Panjab University, Chandigarh 160 014, India
Abstract. Negative index materials (NIMs) are artificially materials which have attracted lot of interest due to their
remarkable properties. In this work, we discuss some of the properties of NIMs and obtained travelling wave solutions for
pulse propagation in NIMs in the presence of external source. The reported solutions are necessarily of the fractional-type
containing trigonometric and hyperbolic functions.
Keywords: Optical solitons, Optical materials
PACS: 42.65.Tg, 42.70-a
INTRODUCTION NIMs, the group and phase velocities always points in
opposite direction.
Negative index materials (NIMs) are designed to have
exotic and unique properties that cannot be obtained with
naturally occurring materials and thus offer entirely new
prospects for manipulating light [1]. In 1967, Veselago
[2] first considered the case of a medium that had permit-
tivity ε < 0 and permeability µ < 0 at a given frequency
and concluded that the medium should then be consid-
ered to have a negative refractive index. In the last few FIGURE 1. Right handed and left handed material
years, theoretical proposals by Pendry et al. [3] for struc-
Let us consider the refraction of a ray at the interface
tured photonic media in certain frequency ranges were
of LHM and RHM. Then as per Snell’s Law, after refrac-
developed experimentally and this has brought Vese-
tion the ray of light turn towards negative side of nor-
lago’s result into the limelight. This field has become
mal in NIMs as shown in Fig. 2. Due to peculiar nature
a hot topic of scientific research and debate over the
Doppler effect will also be reversed in NIMs.
past one decade. In this context study of nonlinear pulse
propagation is new and exciting field of research. M.
Scalora et al. [4] was the first who derived generalized
nonlinear Schrödinger equation (GNLSE) for pulse prop-
agation in NIMs. Various research groups have studied
GNLSE in different contexts and obtained solitary wave
solutions[5, 6]. In this paper we considered GNLSE with
a source term and obtained solitary wave and periodic
solutions. FIGURE 2. Negative refraction and reversed Doppler effect
in NIMs.
PROPERTIES OF NIMS
FRACTIONAL TRANSFORM
It is clear from the Maxwell’s equations that if µ and SOLUTIONS FOR NIMS
ε are simultaneously negative then ⃗ H and ⃗ form
E, ⃗ k
the left handed system as shown in Fig. 1. Hence the The wave propagation in NIMs in the presence of ex-
material is also known as left handed material (LHM). ternal source can be modeled by following equation
The phase velocity of a wave is given by v = ω , hence,
k
(GNLSE)
if k is negative then v should also be negative. We know ∂ ϕ P ∂ 2ϕ ∂ 3ϕ ∂ (|ϕ |2 ϕ )
Poynting vector, ⃗ = 2 (E × H), is always positive, hence
P 1 ⃗ ⃗ i − −iQ 3 + γ |ϕ |2 ϕ +iΛ = β ei(ψ (ξ )−kz) ,
∂z 2 ∂t 2 ∂t ∂t
group velocity will always be positive. Therefore, in (1)

2. where ϕ is complex envelop of the field and P, Q, γ , Λ, β Dark/bright solitary wave solution
represent group velocity dispersion, third order disper-
sion, cubic nonlinearity, self-steepening and external We found general localized solution for the case when
source coefficients respectively. This equation without the Jacobian elliptic modulus m = 1. The set of Eqs.
source has already been studied by Tsitsas et al. [5] and (5) to (8) can be solved consistently for the unknown
obtained solitary wave solutions. parameters A, B, D and for a particular value of c1 . The
In order to find exact travelling wave solutions, we generic profile of the solution reads
have chosen the following ansatz
A + B sech2 ξ
i[ψ (ξ )−kz] α (ξ ) = . (10)
ϕ (z,t) = α (ξ ) e , (2) 1 + D sech2 ξ
where ξ = (t − uz) is the travelling coordinate. Substi- Since the analytical form of solution is known, a simple
tuting Eq. (2) in Eq. (1), and separating real and imagi- maxima-minima analysis can be done to distinguish pa-
nary part we obtain two coupled equations in α and ψ . rameter regimes supporting dark and bright soliton solu-
Choosing ψ ′ (ξ ) = m, the imaginary part can be solved tions. In this case, when AD < B one gets a bright soliton,
to obtain whereas if AD > B then a dark soliton exists. Typical in-
α ′′ = aα + bα 3 + c1 , (3) tensity profile for bright solitary wave solution is shown
2
Λ
where a = − P+u−3Qm , b = Q and c1 is integration con- in Fig. 3b.
Q a b
stant. Using these relations, the real part can be solved
consistently to obtain the various travelling wave param-
( ) 3 2.0
eters as m = 4Λ γ + PΛ , k = aP + Qm3 − um − 3aQm +
1
Intensity
Intensity
2
2Q 2 1.5
2β 1
Pm2
2 and c1 = For β = 0, c1 goes to zero and Eq.
6Qm−P . 0
1.0
1.0 10
1.0
0.5 Z
(3) have bright or dark soliton solutions [5, 6]. 0
0.5 Z 5
0
t t 5
Eq. (3) can be solved for travelling wave solutions by 5 0.0 100.0
using a fractional transformation [7]
FIGURE 3. Typical intensity profiles for (a) periodic and (b)
A + By2 (ξ ) solitary wave solutions.
α (ξ ) = , (4)
1 + Dy2 (ξ )
which maps the solutions of Eq.(1) to the elliptic equa-
tion y′′ ± py ± qy3 = 0, provided AD ̸= B. For explicit- CONCLUSION
ness, we consider the case where y = cn(ξ , m) with m
as modulus parameter. Then upon substitution of Eq. (2) We have obtained periodic and solitary wave solutions
into Eq. (1) and equating the coefficients of equal powers for pulse propagation in NIMs with external source. The
of cn(ξ , m) will yield the following consistency condi- reported solutions are necessarily of the fractional-type.
tions: Negative refractive index provided a new mechanism for
−aA − 2(AD − B)(1 − m) − bA3 − c1 = 0, (5) nonlinear optics in NIMs and resulting in a new class
−2aAD − aB + 6(AD − B)D(1 − m) − 4(AD − B)(2m − 1) of solutions that cannot exist in positive-index materials.
− 3bA2 B − 3c1 D = 0, (6) Because NIMs are artificial materials and we might have
−aAD 2 − 2aBD + 4(AD − B)D(2m − 1) + 6(AD − B)m
the flexibility of controlling these pulses at our will.
− 3bAB2 − 3c1 D2 = 0, (7)
−aBD 2 − 2(AD − B)Dm − bB3 − c D3
1 = 0. (8)
For different values of m, we can obtain different types ACKNOWLEDGMENTS
of travelling wave solutions.
A.G. would like to thank CSIR, New Delhi, for financial
support through S.R.F. during the course of this work.
Periodic solution
REFERENCES
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α (ξ ) = , (9) 3. J. B. Pendry et al., Phys. Rev. Lett. 76, 4773 (1996).
a 1 − 2 cos2 ξ
3
4. M. Scalora et al., Phys. Rev. Lett. 95, 013902 (2005).
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