Download Dispersion Relation of Defect Structure Containing Negative Index

Advance in Electronic and Electric Engineering.
ISSN 2231-1297, Volume 3, Number 8 (2013), pp. 965-970
© Research India Publications
http://www.ripublication.com/aeee.htm
Dispersion Relation of Defect Structure Containing
Negative Index Materials
G.N. Pandey
Department of Applied Physics, Amity Institute of Applied Sciences,
Amity University, Noida (U.P.), India.
Abstract
In this present communication, I have calculated the dispersion relation
and cosine wave of the structure containing negative index materials to
verify transmittance of the considered structure. Such study may be
applicable in telecommunication as filters. To calculate the dispersion
and cosines wave of the materials we have taken Kronig-Penney model
of electronic structure In this addition to this, I have also studied the
optical properties of the defect structure containing negative index
materials.
1. Introduction
Photonic crystals are also known as the electromagnetic wave band gap materials
because the electromagnetic wave cannot propagate through the photonic crystal if the
incident wavelength is equivalent to the thickness of the unit cell of the crystals.
Photonic crystals are the artificial periodic composite materials which exhibit
electromagnetic band gaps [1]
In 1968, Veselago [2] was the first time who proposed a peculiar medium
possesses a negative index (NIM) which contain negative of permittivity (ε) and
permeability (µ) at certain frequency. A NIM or metamaterial causes light to refract, or
bend, differently than in more common positive refractive index materials. Negative
index metamaterials or negative index materials (NIM) are artificial photonic crystal
structures which has negative refractive index over some frequency range [3]. This is
not naturally occurring materials but engineered materials or structures which are
known as metamaterials. Metamaterials which exhibit a negative value for the
refractive index (NIM) are often referred to by any of several names and
terminologies: "left-handed media (LHM), backward wave media (BW media), media
9666
G.N. Pan
ndey
withh negative refractive index, douuble negativ
ve (DNG) metamaterrials, and other
o
simiilar names [4].
[
The photonnic crystal (PC)
(
is a material
m
in which
w
perioddic inclusioons inhibit wave
w
proppagation due
d
to destructive innterference from scatttering from
m the periiodic
repeetition. Thee photonic bandgap
b
prroperty of PCs
P makes them the eelectromagn
netic
anallog of the electronic
e
semi-conducctor crystalss [5]. Intendded materiaal fabricatio
on of
elecctromagneticc band gaaps (EBG) has the goal
g
of creeating periodic, dieleectric
struuctures, withh low loss, and
a that are of high quaality. An EB
BG affects tthe propertiees of
the photon in the same way semicconductor materials
m
afffect the prroperties off the
elecctron. So, it happens thhat the PC is
i the perfecct bandgap material, beecause it alllows
no propagation
p
n of light.
2. Theory and
a Methoodology
By choosing a linearly periodic
p
reffractive ind
dex profile in the perriodic dieleectric
matterial one obbtains a given set of wavelength raanges that are
a allowed or forbiddeen to
passs through thhe periodic dielectric material.
m
Selecting a paarticular x-aaxis through
h the
matterial, I shalll assume a periodic
p
stepp function for
f the indexx of the form
m;
(1))
Where n(x)) =n(x+md)) and m is the translattion factor, which can take the vaalues
m=00, ±1, ±2, ±3,….,
±
and d=d
d 1+d2 is the
t period of
o the latticee with d1andd d2 is being
g the
widdth of the tw
wo regions having
h
refracctive indicess n1 and n2 respectively
r
y. The refracctive
indiices profile of the matterials in thee form of rectangular
r
symmetry iis shown in
n the
figuure (1).[6]
t
periodicc structure the one dim
mensional wave
w
If θ is the angle of inncident on this
equation for thee spatial parrt of the elecctromagnetiic eigen mode is given as,
(2))
Figgure 1: Perioodic refractiive index prrofile of dielectric mateerial.
967
Disppersion Rellation of Def
efect Structuure Containing Negativve Index Maaterials
where n(x) is given byy equation (1).
( Therefo
ore, equationn (2) for waave equation
n for
twoo media mayy be written as,
(3))
(4))
where
a
and
are ray angle in the lay
yer of refr
fractive inddex n1 and
d n2
resppectively.
e
(3
3) ) and (4) can
c be writtten as;
By using Bloch’s theorrem, these equations
(5))
whrre
(6))
On solving above equaation, we obbtain,
(7))
Now, abbreeviating the LHS as
equatio
on (7) may be
b written ass,
(8))
By using equation
e
(8)), we may write the dispersion
d
r
relation
whhich is obtaained
throough the sam
me process calculate
c
thee Kronig-Peenney modeel in periodic potentialss [7].
3. Result an
nd Discussion
In thhis present research
r
paaper I have studied
s
the cosine
c
wavee, band struucture propeerties
of considered
c
s
structure
by KP model and transmiittance of thhe structure is calculateed by
TM
MM. For the calculationns of the opptical properrties of the defect, we have the deefect
matterials of NIM,
N
for thhe considerred structurre containinng differentt values off the
968
G.N. Pandey
cos(Kd)
refractive index of the NIM [6, 8 and 9]. Firstly, we introduce the NIM having
refractive index nd=-1.05. Figure (2) shows the graph between cos(Kd), Band structure
and transmittance versus wavelength (nm) respectively. In this figure cosine wave is
crossing the -1 value for the wavelength range 397nm-403nm. This is exactly verified
the dispersion relation of the considered structure is calculated by using K.P. Model
and the bands structure is the forbidden band within the range of 397nm-403nm which
satisfies the cosine wave. The transmittance graph shows one transmission peak
approximately at 400nm within forbidden band and two transmission peak
approximately at 396nm and 404nm which is out of forbidden range.
-0.8
-1
-1.2
392
394
396
398
400
402
404
406
408
404
406
408
404
406
408
Bandstructure
Wavelength[nm]
6
2.5
x 10
2
1.5
392
394
396
398
400
402
Transmittance
Wavelength[nm]
1
0.5
0
392
394
396
398
400
402
Wavelength[nm]
Figure 2: Cosine wave, band structure and transmittance of structure for
(AB)6DB(AB)2D(BA)6 for nd = -1.05.
Secondly, we are taking the NIM as a defect material having the refractive index nd
= -1.55 for the considered structure which is shown in figure (3) gives the same graph
for cos wave and band structure as given in previous case but this figure gives the
transmission peak at 398nm of same intensity but its value is shifted toward lower
wavelength. From first and second values of the NIM (i.e n = -1.05 and -1.55). We
have the special properties where the tunneling properties is changing as the decreasing
the value of NIM. Such properties show that the periodic structure containing NIM has
tunable properties which can be changed or shifted the wavelength on the value of the
NIM is changed.
Dispersion Relation of Defect Structure Containing Negative Index Materials
969
cos(Kd)
-0.8
-1
-1.2
392
394
396
398
400
402
404
406
408
404
406
408
404
406
408
Bandstructure
Wavelength[nm]
6
2.5
x 10
2
1.5
392
394
396
398
400
402
Transmittance
Wavelength[nm]
1
0.5
0
392
394
396
398
400
402
Wavelength[nm]
Figure 3: Cos wave, bandsstructure and transmittance of structure for
(AB)6DB(AB)2D(BA)6 for nd=-1.55.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
E. Yablonovitch, Phys. Rev .Lett., 58, 2059 (1987).
V.G.Veselago.Sov.Phys.Usp., 10,.509 (1968).
R. A. Shelby, D. R. Smith, S. Shultz, Science, 292, 7779 (2001).
N. Engheta, R. W. Ziolkowski, Physics and Engineering Explorations, Wiley
& Sons, (2006).
W. Chappell, leads the IDEA laboratory at Purdue University
"Metamaterials". Research in various technologies. (2009)
P. Yeh, “Optical Waves in Layered Media”, John Wiley and Sons, USA
(1988)
C. Kittel, “Introduction to Solid State Physics” John Wiley and Sons, USA
(1953).
P.Markos and C.M.Soukoulis, “Wave Propagation from Electron to Photonic
Crystal and Left-Handed materials” (2008)
X. Li, K. Xie and H .M. Jiang Optics Communications 282, 4295 (2009).
970
G.N. Pandey