Download Mofiz U Ahmed

BRAC University Journal, Vol. X, No. 1 & 2, 2013, pp. 13-29
ON TERAHERTZ LASER PULSE PROPAGATION IN
NEGATIVE INDEX METAMATERIALS
U. A. Mofiz1
Plasma Astrophysics Laboratory
Department of Mathematics and Natural Science
BRAC University, 66 Mohakhali
Dhaka-1212, Bangladesh
email: [email protected]
ABSTRACT
The nonlinear propagation of a terahertz laser pulse propagation in the resonant region of a negative
index meta-material is investigated. The resonant frequency of a split ring resonator (SRR),
comprosing the negative index metamaterial, is calculated to be in the order of terahertz. The
analysis shows that the negative refractive index in SRR is maintained within a narrow band of the
frequency, which is more narrowed down with the intensity of the laser radiation. From Maxwell’s
system of equations, a system of coupled equations descring the nonlinear propagation of laser
pulse in negative index metamaterials is obtained. The system of the coupled equations are solved
analytically in the high frequency and in the low frequency responses. In the high frequency
response, a nonlinear field dependent dispersion relation is obtained, and the corresponding group
velocity is determined. In the low frequency response, introducing the Lorentz invariant stretched
coordinates, a system of coupled nonlinear Schrödinger equations (NLSE) are obtained.
Considering the equal intensities of electric and magnetic fields of the laser pulse, the coupled
NLSE equations are solved , which shows that the laser radiation creates bright and dark optical
solitons, respectively, before and after a critical frequency, within the narrow band, determined by
the intensity of the laser radiation. The instability of the bright soliton is investigated while the dark
soliton is modulationally stable.
Key words: meta-material; negative refractive index; bright and dark solitons.
I. INTRODUCTION
I.1. History of Negative Index Metamaterials
Vaselago [1], a Russian physicist in 1967
hypothetically studied the optical properties of an
isotropic medium with simultaneously negative
electric permittivity (  ) and magnetic permeability
(  ), which he named a left- handed material
(LHM). He showed LHMs display unique
"reversed" electromagnetic (EM) properties having



the triad k (wave vector), E (electric field), H
(magnetic field) left handed, hence exhibiting
phase and energy velocities of opposite directions.
A LHM is characterized by negative refractive
index n , therefore, their alternative name, negative
1. For all correspondence
index
materials (NIMs).
Such
materials
demonstrate a number of peculiar properties:
reversal of Snell’s law of refraction, reversal of the
Doppler shift, counterdirected Cherenkov radiation
cone, the refocusing of EM waves from a point
source, etc.
The remarkable property of a LHM is that it
possees a negative refractive index due to which a
thin negative index film will behave as a
“superlens”, providing image detail with a
resolution
beyond
the
diffraction
limit.
Conventional positive index lenses require curved
surface to bent the rays coming from an object to
form an image but in the case of negative refractive
index a planar slab can produce image within the
slab and also outside the slab (Fig.1).
U. A. Mofiz
Negative refraction in photonic crystal is also
feasible [5] in doped graphene metamaterials .
I.2. Negative Refraction in Photonic Crystals
Photonic crystals (PCs) are made only from
dielectrics, thus smaller losses than metalic LHMs,
especially at high frequencies. In PCs, to achieve
negative refraction the size and periodicity of the
“atoms” (the elimentary units) are of the order of
the wave length. In PCs, no effective  and 
can be defined, although the phase and energy
velocity can be opposite as in the case in normal
LHMs. Both negative refraction and superlensing
have observed in PCs [6]. An alternative approach
for fabricating metamaterial is to use PCs
composed of polartonic materials [7].
Fig.1. A negative n medium bends light to a negative
angle relative to the surface normal. Light formerly
diverging from a point source, converged back within the
metamaterial as well as second time in the image plane
outside.
I.3. Graphene Doped Metamaterials
Pendry[2] carefully rexamined the plannar lense
and found that it may recover the evanescent waves
coming from an object. In a planar negative index
lense, an evanesciting wave decaying away from an
object grows exponentially in the lens and deacys
agin until it reaches the image plane (Fig.2).
Graphene is a two-dimensional , one atom thick
allotrope of carbon holds the promise for building
advanced nano-electronic devices since its
discovery in 2004 [8]. It exhibits very unique
optical properties, especially in the terahertz (THz)
frequency range. To date, novel photonic devices
such as THz devices, optical modulators,
photodetectors, and polarizers were successfully
realised. The physics of graphene can be
considered as a unifying bridge between lowenergy condensed matter physics and quantum
field theory , as its two-dimensional quasi-electrons
behaves like massless “relativistic” Dirac fermions,
very
similarly
to
electrically
charged
“neutrinos”[9].
Currently, the huge unexplored potential of
graphene for nonlinear optics has been outlined.
Strong nonlinear optical responses have
investigated in papers [10,11]. Preliminary
experimental include ultrafast saturable absorption
and the observation of strong four wave mixing
[12], which are the building blocks of nonlinear
optics. This discoveries and the advancement in
optical communications are the motivations to
investigate the nonlinear propagation of intense
terahertz laser pulse in meta-materials in the
resonant region.
Fig.2. A decaying wave grows exponentially within the
metamaterial and then decays again utside.
Vaselago’s idea remains hypothetical for a long
time, untill a breakthrough was announced in 2000
by Smith and co-workers [3] presented evidence
for a composite material of conducting rings and
wires – called split ring resonator (SRR)-displaying
the negative value of  and  . The SRR structure
has proven a remarkably efficient means of
producing a magnetic response by scaling down the
size thus upwards in frequency to produce
metamaterials in the terahertz frequencies [4].
In this paper, we have investigated the nonlinear
propagation of terahertz laser pulse in the resonant
region of a negative index meta-material. The laser
pulse is considered as the circularly polarized
electromagnetic (EM) wave. In section II, we have
14
On Terahetrz Laser Pulse Propagation
studied the optical properties of negative index
meta-materials under EM fields and analyzed the
nonlinear resonant frequency of the SRR. In
section III, following Maxwell system equations,
Meta materials based on strong artificial resonant
elements can also be described efficiently with the
Drude-Lorentz formulas. These structures are the
metallic wire structure which provides a
predominently free-elctron respone to EM fields
and the SRR structure provides a predominently
magnetic response to EM fields.
~
~
and the complex magnetic field H are obtained,
coupled equations for the complex electric field E
which are solved analytically in the high frequency
as well as the low frequency responses. In the high
frequency response, a nonlinear field dependent
dispersion relation is obtained, which is analyzed
with the increased field intensities. In the low
frequency response, coupled nonlinear Schrödinger
equations are obtained, the analysis of which shows
that bright and dark optical solitons may propagate
through NIMs within the frequency band, below
and after some critical frequency, which is defined
by the intensities of the wave fields. The bright
soliton is modulationally unstable, its growth rate is
calculated, while the dark soliton is modulationally
stable. Results and discussions are given in section
IV, while section V concludes the paper.
II.2. Nonlinear Form of Dielectric Permittivity
and Magnetic Permeability
The dielectric and the magnetic nonlinear
responses of NIMs under EM fields without losses
are given by the dispersive (frequency dependent)
and the nonlinear (field dependent) expressions as
[14]:
 =  L ( )   NL ; | E |2 ,
 =  L ( )   NL ; | H |2 ,
 L =  0 1   p2 / 2 ,
 L =  0 1  F 2 /  2  02 , .
(6)
 NL =  0 | E |2 /Ec2 ,
(7)
2
2
 NL = 0  | H | /Ec .
(8)
Here,  p is the plasma frequency, F << 1 is the
filling factor,  0 is the linear resonant frequency,
Ec is the critical electric field,  = 1 corresponds
to a focusing dielectric and  = 1 corresponds
to a defocusing dielectric,  > 0 is a fitting
The Drude-Lorentz forms for dielectric permittivity
and magnetic permeability describe the EM
response of materials over the high ranges of
frequencies which are given by [13]:
2


 pe

,
(1)
 = 0 1 2
   02e  ie 


2


 pm

 , (2)
 = 0 1  2
   02m  im 


where,  0 and  0 are the vacuum permittivity
parameter.
II.3. Analysis of the Linear Response
The eigenfrequency of the system of SRRs in the
linear limit is defined as [15]:
 0 0 = 1/c 2 ;
0 = (c/as ) d g /h D 0 ,
 pm are the electron /magnetic plasma
frequency,  0 e and  0 m are the electron /
and
magnetic resonant frequency,
(5)
Under weak fields approximation:
II.1. Drude-Lorentz Form for Linear Dielectric
Permittivity and Magnetic Permeability
 pe
(4)
where,
II. OPTICAL PROPERTIES OF NEGATIVEINDEX MATERIALS UNDER
ELECTROMAGNETIC FIELDS
and permeability, respectively, with
(3)
where,
d g is the size of the SRR gap, h is the
width of the ring,
e and m are the
speed of light.
electron/ magnetic damping coefficient and  is
the frequency of the EM wave. The free electron
contribution in metals corresponds to  0 e  0 .
(9)
a s is the radius and c is the
 D0
is the linear part of the
dielectric. As per Ref. [15] for
 D 0 = 12.8 ,
h = 0.003 cm, d g = 0.1 cm, a s = 0.003 cm,
15
U. A. Mofiz
c = 3 1010 cm s 1 , we find 0 = 9.1  1012 s
1
i.e
the
linear
0
frequency
5
f 0 = 0 /2 = 1.45  10 Hz = 1.45 THz.
For simultaneous  L < 0 ,
L < 0 , and
n =   L L / 0 0
 ,  ,
f . f .
p
p
M
M
, we find
which corresponds
Here,
0 <  <
n
12
min
15
f 0 < f < min
20
0.15
M = 0 / 1  F . For
f M = 1.87 THz; then, 1.45
2
 ,
and
n
band
as functions of frequency
4
0
6
8
10
12
14
0.8
1.0
1.2
1.4
<
1.6
Figure 3: The linear dielectric permittivity  as a
function of frequency  , which remains negative in the
0 <  <  p
0 <  <  M .
| H |2
>= 0.5 , we plot
Ec2
 ,
and
n
as functions of
frequency within the frequency band, which are
shown in FIG.6-8, respectively. The figures show
that the frequency band for NIMs become more
narrower with the applied EM fields.
p
frequency band
0.19
Now, we consider the nonlinear response on the
dielectric permittivity and magnetic permeability, and
on the corresponding refractive index due to applied
EM fields. In this case, for weak fields we have
 2
| E |2 
(10)
 =  0 1  p2   2 ,

E
c 


F 2
| H |2 
(11)
 = 0 1  2
  2 .
2
El 
   0
2
Considering an average value of < | E | >= 0.5 and
Ec2
2
0.6
0.18
II.4. Analysis of the Nonlinear Response
0
0.4
0.17
Figure 5: The linear refractive index n as a function of
frequency  , which remains negative in the frequency
within the frequency band 1.45THz < f < 1.87
THz , which are shown in FIG.3-5, respectively.
0.2
0.16
p
F = 0.4 ; we find
THz < f < 1.87 THz.
Considering
 p = 2 10 THz and 0 = 2 1.45 THz,
we plot
10
.
0
2
0.0
4
0.5
0
6
0
8
1.0
10
1.5
12
14
2.0
0.2
0.16
0.17
0.18
0.19
0.20
0.21
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0.22
p

Figure 6: The nonlinear dielectric permittivity
as a function
of frequency
under EM fields, which remains negative in the
frequency band 0 <  <  P . Solid line (– )line for  = 0 ,

p
Figure 4: The linear magnetic permeability  as a
 , which remains negative in the
frequency band 0 <  < M .
function of frequency
broken line (- - -) for  = 1 (focusing case), dotted line (...) for
 = 1 (defocusing case) .
16
On Terahetrz Laser Pulse Propagation

| H |2 

Ec2 

2
M =
.
| H |2
1  2  F
Ec
H
Introducing H =
, we have
Ec
02 1  
0.0
0
0.5
1.0
1.5
(13)
1/2
2.0
0.16
0.18
0.20
0.22
0.24
For

(14)
as a function
of magnetic field H , which is shown FIG.9. The
figure shows that the magnetic frequency shifts
towards the linear resonant frequency with the
increase of field intensity. It indicates that the
narrow frequency band is more narrowed down
with the increased magnetic field.
p
Figure 7: The nonlinear magnetic permitivity
M  1   | H |2 
=
.
0  1   | H |2  F 
F = .4 and  = 1 , we plot M
as a
function of frequency  under EM fields, which
remains
negative
in
the
frequency
band
0 <  < M . Solid line (– )line for  = 0 ,broken
line (- - -) for  = 1 .
0
2
1.25
6
0
M
4
n
1.20
8
10
12
1.15
14
0.0
0.150
0.155
0.160
0.165
0.170
0.175
0.180
0.2
0.4
0.6
0.8
0.185
H
Figure 9: Magnetic frequency
p
magnetic field intensity | H |
Figure 8: The nonlinear refractive index n as a function
of frequency  under EM fields, which remains
negative in the frequency band
 = 0 and  = 0 , broken line (- - -)
and dotted line (….) for  = 1 ;  = 1 .
M
as
Following Ref. [15], the nonlinear magnetic
permeability can be written as :
 ; | H |  = 1 
2
, then from Eq. (11), we get
FM2
| H |2


= 0,
M2  02
Ec2
 M  0
II.6. Analysis of the Nonlinear Resonant
Frequency
 = M , the magnetic permeability /0 = 0
1
. Here,
as a function of
| H |2
 1.
Ec2
line (– )line for
At
M
Ec2
0 <  < M . Solid
II.5. Analysis of Magnetic Frequency
2
where,
(12)
0 NL
F 2
, (15)
| H |2   2  i
02NL 

is the nonlinear resonant frequency
and  is the loss coefficient. We also have the
which yields
17
U. A. Mofiz
expression for the nonlinear dielectric permittivity
as
 D0
Here,
|E|
.
Ec2
(16)
is the linear dielectric. Then, the
| H | = A
2
1  X  X
2
2
2

X6
  
2
2
1.15
1.10
relation between the macroscopic magnetic field
and the dimensionless nonlinear resonant
frequency can be obtained as :
2 2
0
1.20
oNL
 D | E |2  =  D0  
1.25
2
1.05
1.00
,
0.0
0.2
0.4
0.6
0.8
1.0
H
(17)
c
 = /0 ,
A2 = 16 D3 002 h 2 /c 2 ,
X = 0 NL /0 , and  = /0 .
where,
Figure 11: Nonlinear resonant frequency
function of magnetic field intensity
(defocusing nonlinearity).
|H |
2
0 NL
for
as a
 = 1
Near the resonant region without loss, we consider
 = 1,  = 0 .
These figures (Fig. 10 and Fig. 11) show that the
nonlinear resonant frequency 0 NL goes below the
Then, from Eq.(17), we find
linear resonant frequency
0 NL
0

1
=
2

| H |

2
 A
2
2/3

| H |
  4  
2

 A
2
1/3



III. COUPLED EVOLUTION EQUATIONS
III.1. Maxwell’s Equations
We start with the Maxwell equations


B
 E =  ,
 t
 D
 H =
,
t

  D = 0,

  B = 0,
1.00
0.95
0.90
0
in the case of
focusing (  = 1 ) nonlinearity and it goes up in the
case of defocusing (  = 1 ) nonlinearity.

, (18)


which is shown graphically in FIG.8 and FIG. 9,
for  = 1 (focusing nonlinearity) and  = 1
(defocusing nonlinearity, respectively. From the
available data, we find the value of the resonator
2
parameter A = 2.78 .
oNL
0
0.85

0.80
(19)
(20)
(21)
(22)

where, E and H are the electric and magnetic

0.75
0.70
0.0
0.2
0.4
0.6
0.8
1.0
H
c
Figure 10: Nonlinear resonant frequency
function of magnetic field intensity
(focusing nonlinearity).
|H |
2
0 NL
for

fields, respectively, D and B are the electric and
magnetic flux densities which arise in response to
the electric and magnetic fields inside the medium
and are related to them through the constitutive
relations
as a
given


B = ( L   NL ) H .
 =1
18
by


D = ( L   NL ) E ,
On Terahetrz Laser Pulse Propagation
By using the Maxwell’s systems of equations, we
have the following coupled wave equations
 | E |2
  0  2
t  Ec




2E
2
 E   L  L 2   L 2  NL E  (  E )
t
t

2



 | H | 2  H 
  | H |2  
  0   2  H   0   2 

t  Ec 
 Ec  t 





 NL  H   L  E 
t
t



 L   NL H ,
( 23)
  NL
t 
t

=


 H  L 

H
 L

t

t




   L
zˆ  E  . (26)

t  z





2H
2
 H   L  L 2   L 2  NL H  (  H ) Now, considering that the laser radiation is

t
t
circularly polarized, i. e. E  E x , E y ,0 ,

 


H
 H x , H y ,0 and introducing the complex
=
 NL  E 
 L  H
t
t
~
~
fields: E  E x  iE y , H  H x  iH y , from




  NL  L   NL E.
(24) Eqns. (25) and (26), we find the following coupled
t 
t

equations for the complex fields:

2

Substituting:
 NL = 0  | H |2 /Ec2





 NL =  0 | E |2 /Ec2 ,
and taking



~
~
2E
2E
 2  | E |2 ~ 







E 
L L
0
L
z 2
t 2
t 2  Ec2

2
   | H |  ~
= i 0    2  H 
t  z  Ec  
~
 | H | 2  E  L ~
  0   2  L

E
t  Ec  t
t
  zˆ/z ,
Ez = H z = 0 , the above coupled equations take
the following forms:


2E
2E
 2  | E |2  







E
L L
0
L
z 2
t 2
t 2  Ec2 

    | H |2 
=  0    2  zˆ  H 
t  z  Ec 


 | H | 2  E  L 
  0   2  L

E
t  Ec  t
t

 | E | 2  E 
  | E |2  
  0  2  E   0  2  
t  Ec 
 Ec  t 

  
  L zˆ  H , (25)
t  z

~
 | E | 2  E 
  | E |2  ~
  0  2  E   0  2  
t  Ec 
 Ec  t 
i
   L ~ 
H ,

t  z 
(27)
~
~
2H
2H
 2  | H |2 ~ 








H 
L L
L 0
z 2
t 2
t 2  Ec2

    | E |2  ~
= i 0   2  E 
t  z  Ec  
~
 | E | 2  H  L ~
  0  2  L

H
t  Ec 
t
t


2H
2H
 2  | H |2  








H 
L L
L 0
z 2
t 2
t 2  Ec2


    | E |2 
=  0   2  zˆ  E 
t  z  Ec 

19
U. A. Mofiz
~
 | H | 2  H 
  | H |2  ~




 0   2 H  0   2 

t  Ec 
 Ec  t 
   ~ 
 i  L E,
t  z 
 ( ) = 1 
(28)


1/2
~
H = H(z, t) exp[i(kz  t )] with
(1/ | E |)  | E | /t <<  ,
(1/ | E |)  | E | /z << k ,
 L

 L
and considering
 0,
 0 , L  0,
z
t
t
 L
 0, from Eqn. (27) or Eqn.(28) we find the
z

 | E |2 
 | H |2 
vg = c          2       2 
 Ec 
 Ec 


0 <  <  M
L
and
L
1/2
(29)
frequency
band
for both the linear and nonlinear
0.4
as
0.2
(30)
0.0
0.15
0.16
0.17
0.18
p
(31)
Figure 12: Group velocity

where ,
2
the
c
= c,

1
vg as a function of the
0.8
vg
 0 0
1 
  0  ,
2 
  
2

F  

 L = 0 1  2
 0   ,
    02 


 ( ) = 1 
in
(35)
0.6
1


 L =  0 1 
.
cases, which are shown in FIG.12. The figure
shows that the frequency band for the wave
propagation is more narrowed down (  M
decreases) in the nonlinear case with the increase
of field intensities.
Now, introducing the dimensionless parmeters:
and writing

frequency

1/2
We plot the group velocity

 | E |2 
 | H |2 
 = k  L  L   0  L  2   0 L   2 
 Ec 
 Ec 

with

 | E |2  | H |2 
   2  2 
 Ec  Ec 
the following field dependent nonlinear dispersion
relation:


=
, 0 = 0
p
p

 | E |2 
 | H |2 
      2 
2 
 Ec 
 Ec 
 | E |2  | H |2 
   2  2  .
(34)
 Ec  Ec 
The group velocity vg = /k is found to be
~
E = E( z, t ) exp[i(kz  t )] and
.
 
 = kc        
III.2. High Frequency Response and Nonlinear
Dispersion Relation
 | E |2  | H |2 
  0 0  2  2 
 Ec  Ec 
(33)
the above dispersion relation (Eq.(29)) can also be
written in the following form:
which we study below for the high frequency and
the low frequency responses to the fields.
Taking
F 2
 2  02
in the frequency band
vg
as a function of frequency
0 <  < M . Solid line
(—-) is the linear case, dash and dot lines(- - -) for
(32)
nonlinear
case
for
 = 1 , respectively.
20
 = 1,
| E |2
| H |2
= .5, 2 = .5
2
Ec
Ec
and
On Terahetrz Laser Pulse Propagation
~
~
2H
2H
   
Z 2
T 2
~
~
 2 | H |2 ~
~ 2 2H
  
H    | H |
T 2
T 2
~
~
~
   2 | E |2
 | E |2 E 
= i 0 
Ex 

0  TZ
Z T 
~ 2
~
~
 | E
|  | H |2 ~ ~ 2  2 | H |2 ~
  
H| E |
H
T
T 2
 T
~
~
~  | H |2 H 
 | E |2

T T 
~ 2 ~
~
 | E
| H
~ 2 2H 
   
|E |

T 2 
 T T
~ 2 ~
~
~
 | E
| H ~ 2
~ 2  | H |2 H
  
|H | |E |
T T
 T T
~
~ ~ 2H 
 | E |2 | H |2

T 2 
~
 0   2   ~   E 
i
E

. (37)
 0  TZ
Z T 

III.3. Low Frequency Response: Coupled NLS
Equations
Normalizing the field variables

E and H by Ec
and
introducing
dimensionless
space-time
coordinates: Z = z/ p , and T = t/t p , where  p
is the pulse length, and
 p  ct p ,
t p is the pulse time with
the coupled equations, Eqs. (27) and
(28) in dimensionless forms are:
~
~
2E
2E
   
Z 2
T 2


~
~
 2 | E |2 ~
~ 2 2E
  
E    | E |
T 2
T 2


~ 2
2
~
~
 0   | H | ~  | H | 2 H 
= i
H


 0  TZ
Z T 

~
~
~
  | H |2  | E |2 ~ ~ 2  2 | E |2 ~
  
E | H |
E
T
T 2
 T

Now, we consider the following Lorentz invariant
stretched coordinates [16]:  =  Z  vg T and


  1  v 
~
~
~ 2  | E | 2 E 
| H |

T T 
~
~
~
  | H | 2 E
~ 2 2E 
   
|H |

T 2 
 T T
 =  2 T  vg Z  , where
 < 1 , vg  vg /c , and we expand

2 1/2
,
g
the quantities
~
~
E and H as:

~
E =  2E0   l El eil ( kZT )  E*l e il ( kZT ) , (38)
~
~
 ( )   | H | 2 ~
~ 2 E 

E 2| H |


T  T
T 
 2 ( ) ~ 2 ~

|H| E
T 2
~
~ 2 ~
~
 | H
| E ~ 2
~ 2  | E | 2 E
  
|E| |H |
T T
 T T
~
~ ~ 2E 
 | H |2 | E |2

T 2 
~
 0   2   ~   H 
i
H

, (36)
 0  TZ
Z T 


l =1

~
H =  2H0   l Hl eil ( kZT )  H*l e il ( kZT ) . (39)
l =1
Here,
(E0 , H0 , El , Hl )  A(1)  A(2)   2 A(3)  ...
are functions of stretched coordinates ( , ) .
Then in the order 0( ) , we find  = k/   , and
in the order 0( 2 ) , we find vg = 1/   approving
the compatibility condition. Thus, following the

standard technique [17] in the order 0( ) , we
arrive at the following coupled NLS equations:
3
and
21
U. A. Mofiz
a
 2a
 P 2  Qa | a |2 a  Qb | b |2 a = 0, (40)


b
 2b
i
 P 2  Qb | b |2 b  Qa | a |2 b = 0, (41)


0.000
i
Qa
0.002
a  E1(1) , b  H1(1) and P , Qa , Qb are
where
0.006
given by the expressions



0.008
 1   vg2
P=
,
2    kvg /

 
,
2    kvg /

Qa =

0.150
0.175
0.180
0.185

nonlinearity)
0.008
0.006
Qa
are the nonlinear coefficients, the nature of which
on the frequency  in the frequency band
0 <  < M , we study below.
0.004
0.002
P , Qa , and Qb
0.000
0.150
0.155
0.160
0.165
F = 0.4 , 0 = 0.145 ,
0.170
0.175
0.180
0.185
p
<| a | >= 0.5 , <| b | >= 0.5 ,  = 1 , and
 = 1 , the plots of P , Qa , Qb , and
Q = Qa  Qb are depicted in FIGs. 13-17,
2
2
0.170
in the frequency band 0 <  < M . Qa
is always negative but changes its value abruptly at a
critical frequency cr  0.1742 for  = 1 (focusing
frequency
(43)
P is the dispersion coefficient, Qa and Qb
Taking the parameters
0.165
Figure 14: The nonlinear coefficient Qa as a function of

III.4. Analysis of
0.160
p
 
Qb =
. (44)
2    kvg /
Here,
0.155
(42)
and

0.004
Figure 15: The nonlinear coefficient Qa as a function of
frequency  in the frequency band 0 <  < M . Qa
is always negative but changes its value abruptly at a
critical frequency cr  0.1742 for  = 1
respectively.
(defocusing nonlinearity)
0.0
4
0.5
P
Qb
2
1.0
0
1.5
2
2.0
0.150
0.150
0.155
0.160
0.165
0.170
0.175
0.180
0.155
0.160
0.165
0.170
0.175
0.180
0.185
0.185
p
p
Figure 16: The nonlinear coefficient Qb as a function of
Figure 13: Dispersion coefficient P as a function of
frequency  in the frequency band 0 <  < M . P

in the frequency band 0 <  < M . Qb
is always negative but changes its value abruptly at a
critical frequency cr  0.1742 .
frequency
changes its sign from negative to positive at a critical
frequency cr  0.1742 .
22
On Terahetrz Laser Pulse Propagation
0.0
2
0.5
Q
2
magnetic fields: X = Y , we plot the graph of
cr as a function of field X = Y , which is
shown in FIG.18. The figure shows that the critical
frequency  cr goes to the linear resonant
1.0
frequency
0
with the increase of intensities of the
wave fields.
1.5
0.185
2.0
0.150
0.155
0.160
0.165
0.170
0.175
0.180
0.185
0.180
p
cri
0.175
Figure 17: The nonlinear coefficient
as a function of
0 <  <  M .
0.170
is always negative but changes its value abruptly at a
0.165
frequency
Q
Q

in the frequency band
critical frequency
cr  0.1742 .
0.160
0.0
We see from the Figs. 13-17 that P changes its
sign from negative to positive , Qa is negative for
0.4
0.6
0.8
1.0
X
cr as a function of applied field X = Y .
Note that cr  0 as X  1 .
Figure 18:
focusing ( 
= 1 ) nonlinearity while it is positive
for defocusing(  = 1 ) nonlinearity , Qb an Q
are always negative but change their values
abruptly at a certain critical frequency  cr , which
is defined by the condition at
0.2
III.5. Solution of the NLSE
From the symmetric nature of Eqs. (40) and (41)
and considering the same intensities of the electric
and magnetic fields of the wave, we can take
vg /c = 1 , and it is
determined from the Eq.(35) as
| a |2 =| b |2 [18]. In this situation, we obtain from
cr =
 bˆ  bˆ 2  2aˆcˆ
,
2aˆ
either of Eq. (40) or Eq. (41)
a
 2a
 P 2  Q | a |2 a = 0,
(46)


where, Q = Qa  Qb . In the present situation, it is
seen that, P can be positive as well as negative for
(45)
i
where,
aˆ =  F  Y 2  (1  F )X 2  X 2Y 2 ,
different range of the values of the frequency  ,
whereas Q is always negative. Thus, for obtaining
solution of the above Eq. (46), we may have two
bˆ = (1 F )  02 (X 2  Y 2  X 2Y 2 )  Y 2 ,
cˆ = 02 (1  Y 2 ). (37)
|E|
| H|
Here, X =
and Y =
.
Ec
Ec
cases: Case I:
Q < 0 , and
 0 <  <  cr ;
 cr <  <  M ;
For
PQ > 0
where
where P < 0 ,
and
Case
II:
P > 0 , Q < 0 , and
 = 1, X = 0.5, Y = 0.5, 0 = 0.145, F = 0.4 ,
we find cr = 0.174822 for  = 1 , and
cr = 0.174837 for  = 1 , respectively.
PQ < 0 . For the case I, we take a new variable 
such
that
and
assume
 =   V
a( , ) = u( ) exp[i( K   )] , where V is
Considering the same intensities of electric and
a constant, then from Eq. (46) we find:
23
U. A. Mofiz
(2PK  V )
P
du
= 0,
d

a0 , V and D are arbitrary constants. They
define gray solitons . For V = 2PD , it is a kink
Here,
(47)

d 2u
 PK 2   u  Qu 3 = 0,
d 2
type dark soliton. Below, we investigate the
modulational instabilities of these solitons.
(48)
III.6. Modulations instabilities of optical solitons
which gives
motion
(
K = V/2P and yields an integral of
Coherent dispersion: Hasegawa Formalism
du 2  PK 2    2 Q 4
u 
)  
u = const. (49)
d
P
2P


For
a
localized
solution,
we
In order to investigate the modulational instability
of optical solitons, we follow Hasegwawa
formalism [20,21]. Considering the modulation of a
constant amplitude coherent phase dispersion, we
consider
u, du/  0 as    , and so, we put
const. = 0 . At the maximum value of u = u0 ,
which
defines
du/d = 0 ,
introduce a phase shift
a  aExp(iQ | a0 |2  ) ,
a 0 is the constant equilibrium amplitude of
where
the field. Then, Eq.(46) can be written as:
 = Qu02 /2  V 2 /4P .
i
Thus, we arrive at the following solution of Eq.(49)
a
 2a
 P 2  Q(| a |2  | a0 |2 )a = 0. (53)



Q 
(50)
u( ) = a0 sech a0
 ,
2P 

where a 0 is a constant. Going back to the ( , )
In
a = Exp(i ) , where  =  ( , ) and
 =  ( , ) .This means that  =| a | with an
equilibrium value  0 =| a0 | . Equation(53) has the
polar
representation,
coordinates, we have the solution of Eq. (46) as
obvious solution
 =  0 , = 0 .


Q
a( , ) = a0 secha0
(  V )
2P


e

1
V 2  
V

i
   Qa02 


 
2
P
2
2
P


 


i
e
we
write
 =  0  1 , where | 1 |<<  0 and
 =  1 . Then by linearizing Eq.(53) from the
Let
imaginary and real parts , we obtain
,
(51)
which is a bump type solution, known as the bright
soliton, and the wave amplitude may be
modulationally unstable. Similarly, the solution for
Case II ( PQ < 0 ) is [19]:


a( , ) = a0 tanha0


a

Q
(  V )
2P

(54)
 1 P  2 1

 2Q0 1 = 0.
 0  2
(55)
which yield
4
 2 1
2  1

P
 2QP02 1 = 0.
(56)
 2
 4
Considering,
1 = 1Exp(iK   ) ,we
2P 
V 
D

Q 
2 P 



V 2  
i  D  u02Q 3 PD 2  2 DV   

2P  


 
1
 2 1
 P0
= 0.

 2
readily obtain from
dispersion relation
Eq.56),
the
nonlinear
 2 = PK 2 ( PK 2  2Q 02 ) = P 2 K 2 ( K 2 
,
= ( PK 2  Q02 ) 2  Q 2 04 .
(52)
24
2Q 2
0 )
P
(57)
On Terahetrz Laser Pulse Propagation
We see from Eq.(57) that if Q/P > 0 ,
becomes negative for values of
2
A. Fast Mode: Bright Soliton
K below
We see from above, the fast mode has PQ > 0 ,
and thus modulationally unstable. Possible final
state can lead to the formation of bright soliton, i.e.
a localized envelope modulating a carrier wave in
the form a = Exp(i ) , where
2Q
K cr =
| 0 | , so that there is a purely growing
P
mode and the wave is modulationally unstable. The
growth rate  = Im() attains a maximum
 max = Q | 0 |2
Q
|  0 | . Therefore, in
P
dimensionless variables:  = / max , K = K/K cr :
for K =
 = 1  (2K 2  1) 2 .
=
(58)
(59)
and

FIG.19 shows the variation of the growth rate
1 2P
   V 
sech
.
L Q
 L 
=
as a function of the wave number K . On the other
hand, for Q/P < 0 , the wave is modulationally
stable.
where,
 2P V 2  
1 


V



 L  2    0 . (60)
2 P 

 
V is the envelope speed, L is the pulse
 = 0 and  0 is an arbitrary
spatial width at
It is to be noted that for a fixed amplitude  0 , the
critical frequency K cr is a function of frequencies
phase. The maximum amplitude
 in the frequency band, which is shown in FIG.
0
is inversely
L , i.e. 0 L = 2P/Q .
We do an analysis of pulse width L on the
proportional to the width
20.
1.0
frequencies within the band, which is shown in Fig.
21. It shows that the width is narrower at the
resonant frequency while it becomes wider at
higher frequencies.
0.8
max
0.6
0.4
3.0
0.2
2.5
0.0
0.2
0.4
0.6
0.8
2.0
1.0
L
0.0
k
Figure 19: The growth rate

1.5
k cr
as a function of
K
.
1.0
0.5
6
0.150
5
0.155
0.160
0.165
0.170
k cri
4
p
3
Figure 21: Pulse width L as a function frequency
within the frequency band for  0 = 1 .
2

1
At a fixed time  = 0 , we have shown
modulational envelopes of the carrieer wave and
the corresponding bright solitons at different fixed
frequencies within the frequency band in FIGs. 2223 for L = .5 and FIGs. 24-25 for L = 2 ,
respectively.
0
0.150
0.155
0.160
0.165
0.170
p
Figure 20: K cr as a function frequency  in the
frequency band for  0 = 1 .
25
U. A. Mofiz
1.0
1.0
0.9
0.5
2
0
I I2
a
a0
0.8
0.0
0.7
0.6
0.5
0.5
1.0
2
1
0
1
2
0.4
2
L
1
Figure 22: Modulational envelope of fast mode for
L = .5 corresponding to / p = .155 , V/2P = 30 at
1
2
L
 =0.
Figure 25: Bright soliton for
/ p = .17 , at  = 0 .
1.0
L=2
corresponding to
B. The slow mode: Dark Soliton
0.8
The slow mode is modulationally stable since
PQ < 0 . It produces dark and gray solitons[22].
The dark has
2
0
0.6
I I2
0
0.4
=
0.2
0.0
2
1
0
1
1 2P
   V 
tanh
.
L Q
 L 
(61)
1 
V2 
V     0 ,
2P 
2 
(62)
2
and
2.
Figure 23: Bright soliton for L = .5 corresponding to
/ p = .155 , at  = 0 .
=
1.0
while the gray has
0.5
a
a0
= |
0.0
2P 1
   V
|
1  d 2 sech2 
Q Ld
 L
0.5
and
1.0
V02 
1 

=
V0 

2 P 
2 
2
1
0
1
2

 . (63)

   V 
(64)
d  tanh

L 

s  arcsin
 0 .
2
2    V 
1  d sech 

 L 
L
Figure 24: Modulational envelope of fast mode for
L = 2 , corresponding to / p = .17 , V/2P = 30 at
 =0.
26
On Terahetrz Laser Pulse Propagation
Here, the parameter d, lying in the range
0 < d < 1, regulates the modulation depth,
0.6
0.4
s = 1 , and V0 = V  s(2P/Ld ) 1  d 2 . Note
that for d = 1 , one recovers the dark soliton. The
a
a0
0
finite equilibrium amplitude
0.2
is now inversely
0.2
L and the
parameter d , i.e. 0 Ld = 2 | P/Q | . The
proportional to both the width
minimum
amplitude
 min =  0 1  d
2
is
given
0.0
0.4
0.6
by
2
1
0
1
2
, which is zero in the dark
L
case. FIGs 26-29 show the modulational envelopes
of the carrieer wave and the corresponding dark
solitons at different frequencies within the
frequency band at a fixed time  = 0 for L = .5
and L = 2 , respectively.
Figure 28: Modulational envelope for slow mode
corresponding
L = 2,
V/2P = 30 at  = 0 .
to
/ p = .17 ,
0.5
1.0
I I2
2
0
0.4
0.5
0.3
a
a0
0.2
0.0
0.1
0.0
0.5
2
1
1.0
0
1
2
L
2
1
0
1
2
Figure 29: Dark soliton for
/ p = .17 , at  = 0 .
L
L=2
corresponding to
Figure 26: Modulational envelope of slow mode for
L = .5 corresponding / p = .155 , V/2P = 30
at  = 0 .
IV. RESULTS AND DISCUSSIONS
Thus, we analyze the optical properties of the
negative index materials (NIMs) under external
electromagnetic (EM) fields of intense laser
radiation, their linear and nonlinear resonant
frequencies, and the propagation of EM fields
through NIMs near the resonant region. With the
data available, the linear resonant frequency
(eigenfrequency) of SSR is found to be 1.45 THz
and the magnetic frequency to be 1.87 THz. The
resonant frequency and the magnetic frequency
creates the frequency band within which the
dielectric permittivity and the
magnetic
permeability simultaneously are negative, which is
the main criteria of NIMs. Thus, the founded
frequency band is a narrow band. Analyzing
Eq.(10) and Eq.(11), it is found that the frequency
band becomes more narrower with the increasing
intensities of the EM fields of the propagating
1.0
0.8
2
0
I I2
0.6
0.4
0.2
0.0
2
1
0
1
2
L
Figure 27: Dark soliton for
/ p = .155 , at  = 0 .
L = .5
corresponding to
27
U. A. Mofiz
wave. Eq.(14) shows that that the magnetic
frequency shifts towards the linear resonant
frequency with the increase of field intensity. We
solve Eq.(17) to find the relation between the
nonlinear resonant frequency with the intensity of
the wave’s magnetic field. With the increasing
magnetic field intensity, Eq.(18) shows that the
nonlinear resonant frequency goes below the linear
resonant frequency in the case of focusing
nonlinearity, while it goes up in the case of
defocusing nonlinearity . A system of coupled
equations [ Eq.(25) and Eq.(26)] describes the EM
fields propagation through NIMS. From these
equations, in the high frequency response, we find
the the nonlinear field dependent dispersion
relation defined by the Eq. (34). The group velocity
of the corresponding wave is defined by the Eq.
(35) which shows that the frequency band for wave
propagation is more narrowed down with the
increase of the field intensities.
wave. The nonlinear resonant frequency of the split
ring resonator depends on the intensity of the wave,
and it goes below the linear resonant frequency in
the case of focusing nonlinearity, while it goes up
in the case of defocusing nonlinearity. The
nonlinear field dependent dispersion relation and
the corresponding group velocity determined in the
high frequency response, show that the frequency
band of the wave propagation is more narrowed
down with the increase of field intensities. The
coupled
nonlinear
Schrödinger
equations
describing the propagation of the wave in the slow
response, predicts the propagation of bright
solitons and dark solitons before and after a critical
frequency determined by the field intensities within
the frequency band. The bright soliton is a fast
mode and it is modulationally unstable. The growth
rate of the instability is calculated. The dark soliton
is a slow mode and it is modulationally stable.
Solitons pulse widths are more narrower at the
resonant frequency and they become more wider
away from the resonant region. Results obtained in
this investigation may have some applications in
meta-materials using SRR with the negative
refractive index in the terahertz spectrum and in
photonic crystals, which are to be investigated in
near future.
In the low frequency response, using the reductive
pertubation method, we obtain a coupled [(Eq.(40)
and Eq.(41)] nonlinear Schrödinger (NLS)
equations . The dispersion coefficient and the
nonlinear coefficients of the NLS equations are
investigated within the frequency band of NIMs.
Considering the same intensities of electric and
magnetic fields of the wave, the coupled system is
solved, which predicts the propagation of bright
and dark solitons through NIMs before and after
some critical frequency within the band. Eq.(45)
defines the critical frequency, the analysis of which
shows that it goes to the linear resonant frequency
with the increase of field intensities. The bright
soliton is a fast mode which is modulationally
unstable. The growth rate of the instability is
calculated and it is shown by Eq.(58). Solitons
pulse widths are also investigated within the
frequency band, which shows that they are more
narrower at the resonant frequency and they
become wider away from the resonant region. The
dark soliton is a slow mode which is
modulationally stable.
ACKNOWLEDGEMENT
This work has been supported by the Ministry of
Education of the Government of Bangladesh under
Grants for Advanced Research in Science:
MOE.ARS.PS.2011. No.-86. Thanks to Professor
Mohammad Zahidur Rahman for encouraging the
publication of the research paper on advanced
materials in BRAC University Journal.
REFERENCES
V. CONCLUSION
To summarize, we have investigated the nonlinear
propagation of a terahertz laser radiation in the
resonant region of a negative index meta-materials.
It is found that the negative refractive index of the
meta-material is maintained within a narrow band
of frequencies, which is more narrowed down with
the increase of field intensities of the propagating
28
1.
V. G. Vaselago, The electrodynamics of
substances with simultaneously negative
values of  ,  . Sov. Phys. Usp 10, 509,
1968, pp. 14-21 [Uspekhi Phis. Nauk, 92 (3),
1967, pp. 519-526].
2.
J. B. Pendry, Negative refraction makes a
perfect lens, Phys. Rev. Lett, 85,2000, pp.
3966-9.
3.
D. R. Smith, W. J. Padilla, D. C. Vier, S. C.
Nemat-Nasser, S. Schultz, Composite medium
with simultaneously negative permeability
and permittivity, Phys. Rev. Lett., 84, 2000,
pp. 4184-4189.
On Terahetrz Laser Pulse Propagation
4.
S. Linden, C. Enkirch, M. Wegner, J. Zhou,
T. Koschny, C. M. Soukoulis, Magnetic
response of metamaterials at 100 terahertz,
Science, 306, 2004, pp. 1351-1353.
5.
P. V. Parimi, W. T. Lu, P. Vodo, S. Sridhav,
Photonic crystal: Imaging by flat lense using
negative refraction, Nature, 426, 2003, p.
404.
6.
14. I. Kourakis and P. K. Shukla,
propagation of electromagnetic
negative
refraction
index
materials, Phys. Rev. E 72,
0166261-1 - 0166261-4.
Nonlinear
waves in
composite
2005, pp.
15. A. A. Zharov, I. V. Shadrivov and Y.
Kivshar, Nonlinear properties of left-handed
metamaterials, Phys. Rev. Let., 91, 2003, pp.
037401-1 - 037401-4.
E. Cabuku, K. Aydin, E. Ozbay, S.
Foteinopoulu,
C.
M.
Soukoulis,
Subwavelength resolution in a twodimensional
photonic
crystal
based
superlens, Phys. Rev. Lett, 91, pp. 207401-1 207401-4.
16. G. I. Melikidze, J. A Gil, A. D. Patarya, The
spark associated soliton model for pulsar
radio emission, Astrophys. J. 544, 2000,
pp.1081-1097.
7.
K. C. Huang, M. C. Povinelli, J. D.
Joannopoulos, Negative effective permiability
in polaritonic photonic crystals, Appl Phys.
Lett., 85, 2004, pp. 543-545.
17. U. A. Mofiz and M. R. Amin, Langmuir dark
solitons in dense ultrarelativistic electronpositron
gravito
plasma
in
pulsar
magnetosphere. Astrophys. Space Sci. 345,
2013. pp.119-124.
8.
K. S. Novoselov, A. K. Gleim, S. V.
Morozov, D. Jiang, Y. Zang, S. V. Dubonos,
I. V. Grigorieva and A. A. Firsov, Electric
Field effect in atomically thin carbon films,
Science, 306, 2004, pp. 606-669.
18. N. Lazarides and G. P. Tsironis, Coupled
nonlinear Schrödinger field equations for
electromagnetic wave propagation in
nonlinear left-handed materials. Phys. Rev.
E, 71, 2005, pp. 036614-1-4.
9.
A. K. Geim, Graphene: Status and Prospects,
Science, 324, 2009, pp. 1530-1534.
19. C. L. Nam, Derivation of nonlinear
Schrödinger equation for electrostatic and
electromagnetic waves in fully relativistic
two-fluid plasmas by the reductive
perturbation method. Phys. Plasmas 19, 2012,
pp. 082303-1 - 082303-13.
10. A. R. Wright, X. G. Xu, J. C. Cao and C.
Zhang, Strong nonlinear optical response of
graphene in the terhertz regime, Appl. Phys.
Lett. 95, 2009, pp. 072101-1 - 072101-3.
20. A. Hasegawa, Plasma Instabilities and
Nonlinear Effects (Springer, Berlin), 1975 .
11. K. L. Ishikawa, Nonlinear optical response of
graphene in time domain, Phys. Rev. B 82,
2010, pp. 201402-1 - 201402-4 (R).
21. U. A. Mofiz, On the generation of large
amplitude spiky soliton by ultralow frequency
earthquake emission in the Van Allen
radiation belt, Phys. Plasmas, 13, 2006, pp.
082901-1 - 082901 -10.
12. E. Hendry, P. J. Hale, J. Moger, A. K.
Savchenko and S. A. Mikhailov, Coherent
nonlinear optical response of graphene, Phys.
Rev. Lett. 105, 2010, pp. 097401-1 - 0974014.
22. T. Cattert, I. Kurakis and P. K. Shukla,
Envelope
solitons
associated
with
electromagnetic waves in a magnetized pair
plasma. Phys. Plasmas 12, 2005, pp. 0123191 - 012319-6.
13. C. M. Soukoulis, M. Kafesaki, E. N.
Economou, Negative Index Materials: New
Frontiers in Optics, Adv. Mater, 18, 2006,
pp. 1941-1952.
29