Download Reflection properties, anomalous group velocity and negative

Reflection properties, anomalous group velocity and negative
refractive index in one-dimensional Plasma Photonic Crystal
Khem B. Thapa, S. K. Singh, S. K. Srivastava, Om Prakash, G. N. Pandey, U. N. Singh and
S. P. Ojha *
Department of Applied Physics, Institute of Technology,
Banaras Hindu University, Varanasi-221005, INDIA
ABSTRACT
The present paper describes the study of reflection of electromagnetic waves in the far ultraviolet and visible region of
electromagnetic (EM) spectrum in one-dimensional Plasma Photonic Crystals (PPCs). The Plasma Photonic Crystal
structure is composed of alternate layers of thin plasma and dielectric material. In order to obtain the characteristic
equation (dispersion relation), we have solved the Maxwell scalar wave equation. Anomalous behavior of group velocity
has been observed. Occurrence of negative effective refractive index in Plasma Photonic Crystal (PPC) is worth noting.
Also near the photonic band edge the effective index of refraction (neff) attains exceptionally high values greater than the
refractive indices of the materials n1 and n2 used. Moreover, the effect of change in the plasma width on the reflectivity
of proposed PPC structure has been studied. It has been found that the 100% reflectivity region of the PPC structure
increases as the width of plasma material increases.
Key words: Plasma Photonic Crystal, effective refractive index, anomalous dispersion, negative refractive index,
negative group velocity.
1. INTRODUCTION
In recent years, Photonic Crystal structures are the subject of intensive and widespread study because they can control
the propagation of electromagnetic waves in a ways that conventional optics cannot [1-3]. Photonic Crystal, also known
as Photonic Band Gap Material (PBG), are periodic arrangement of dielectric or metallic elements with high refractive
index contrast in one-, two-, and three-dimensions. PBG structures are characterized by the presence of well defined
forbidden frequency bands (photonic band gap), where the propagation of electromagnetic waves is prohibited. Inside
the photonic band gap mode propagation, spontaneous emission and zero point fluctuations are all absent. This ability of
photonic crystals to control the light with amazing way has numerous applications in optical filters, microcavities,
thresholdless semiconductor lasers, frequency selective surfaces, photonic crystal fibers and microwave antennas [4-8].
Moreover, these structures can be engineered for a different frequency range simply by scaling their dimensions [9]. The
photonic devices have greater advantages over conventional electronics devices, because they offer very high operating
speed, greater lifetime, tolerance to temperature fluctuations and capability to care high repetition rates. All these devices
work on the principle of photonic band edges and they are extremely compact in nature.
Recently, Mase et.al. [10] has shown that the frequency gap in a periodic array composed of alternating thin micro
plasma and dielectric materials becomes larger with the increase of plasma density as well as plasma width. Fukaya et.al.
[11] studied optical transmission properties of the periodic multilayers of two different materials.
In this paper, we have reported two types of studies. In its first part, we have calculated the reflectivity of the Plasma
Photonic Crystal (PPC) structure having plasma and dielectric materials in their alternate regions in the far ultraviolet
and visible region of the electromagnetic spectrum. The reflectivity is calculated for a given set of incident angles and it
is found that the proposed structure reflects the maximum portion of far ultraviolet and visible spectrum. Moreover, the
effect of variation of plasma thickness on the reflectivity of the proposed structure has also been observed. In the second
part, an interesting study have been carried out wherein effective refractive index becomes negative and attains ultrahigh
value at the band edges. In order to calculate the expression for effective refractive index inside the PPC structure we
have used the concept of group velocity, instead of phase velocity as suggested by Sakoda [12]: “The phase velocity is
defined as the velocity of propagation of an equi-phase surface. This velocity has a definite meaning, for example, for
plane waves and spherical waves for which equi-phase surface can be defined without ambiguity. In the photonic
crystals, however, the equi-phase surface can not be defined rigorously, since its eigen function is a superposition of
plane waves. This means that the phase velocity can not be defined appropriately in photonic crystal. Recently,
D’Aguanno et.al. [13], Jeong et.al. [14], Woodley et. al. [15], Ojha et.al. [16] have also calculated the group index by the
same formula as suggested by Sakoda.
A further justification in favour of the definition of effective index of refraction (neff((g)) in terms of the group velocity
is that all light sources in actual practice even monochromatic ones are wave packets and has some, though extremely
small, frequency spread. Defining neff(g) in terms of the group velocity leads to some very interesting results. For
example, the effective index of refraction becomes negative in the certain range of frequencies .In recent time these
artificial materials with the so-called negative index of refraction in a certain range of frequencies have generated much
scientific interest [17-18].
The possibility of negative refractive index was first raised by the Russian physicist, Victor Veselago, [19] in the mid1960s. He predicted that if the electric permittivity and the magnetic permeability of the material were both negative
then the Maxwell’s equations of electromagnetism would give a negative refractive index. By adopting the concept of
group velocity some useful and interesting results are obtained which shows a myriad potential application of photonic
band gap materials in photonic technology apart from meeting the requirement for lasing without inversion as suggested
by Scully [20] and discussed by Dowling and Bowden [21].
In the following section we shall derive the analytical expressions for the dispersion relations, reflectivity, group
velocity (Vg), effective refractive index neff(g) and the effective index of refraction(neff(p)) in a one-dimensional Plasma
Photonic Crystals .
2. THEORETICAL ANALYSIS
The Maxwell’s equation for EM wave propagating along the x-axis may be written as [22, 23]
d 2 E( x)
dx 2
with
2
 ω



+  n(x ) − β 2  E ( x) = 0
 c




β=
ω
n ( x) sin(θi )
c
(1)
i = 1,2
where β is propagation wave vector constant, c is velocity of light and n(x) is the periodic refractive index profile of the
structure given by
n , b <x <d
n ( x) =  1
 n 2, 0 < x < b
 ω

p
with n (x ) = n(x + d ) where, n1 is the refractive index of the dielectric material and n 2 = 1 − 
  ω

(2)
1
2 2








is the refractive
1
 e2 n  2
p
index of the plasma [10]. Here, ω is the frequency of electromagnetic wave and ω p = 
is the plasma
 ε m 
0


frequency with np as electron density, d = a + b is the period of the lattice with a and b being the width of the dielectric
and plasma layers respectively. The geometry of the structure is shown in figure 1.
The electric field E( x ) within each homogeneous layer is a combination of right-traveling waves and left-traveling
waves and so it can be expressed as the sum of an incident plane wave and reflected plane wave. Thus, the electric field
in the mth unit cell can be written as
a e−ik1 (x − md) + b eik 2 ( x −md ); ( md − a ) < x < md
 m
m
E( x ) = 
−
ik
(
x
−
md
+
a
)
2
 cm e
+ dm eik 2 ( x −md +a ); ( m − 1)d < x < ( md − a )
,
(3)
where
1 2
 ω  2

k1 =  n1  − β2 
 c 



n ω
= 1 cos θ1
c
(4a)
1 2
 ω 2

n ω
and
(4b)
k 2 =  n 2  − β 2 
= 2 cos θ 2
c
 c




Here, am, bm, cm and dm are constants and θ1 and θ 2 are the ray angles in the layers and they are related with the
incident angle θ1 by the following relation,
1
  2
 2
n
(5)
θ2 = cos−1 1 −  1  sin 2 θ1 
  n2 



dE(x )
Now imposing the continuity of E(x) and
at the interfaces x = (m − 1) d and x = (m − 1)d + b we get four equations
dx
having four unknown constants. Solving these equations, we get the following matrix equation
a m −1 A B a m 
  
b
= 
 m −1 C D  bm 
(6)
where A, B, C and D are the matrix elements, given as



1 1
A = e ik1a  cos k 2 b + i + γ  sin k 2 b 
2 γ







1 1
B = e− ik1a  i − γ  sin k 2b

2  γ

(7a)
(7b)
 1 1


C = eik1a  − i − γ  sin k 2b

 2 γ




1 1
D = e− ik1a  cos k 2b − i + γ  sin k 2b
2 γ



(7c)
(7d)
k
k n2
where γ e = 1 for TE mode and γ m = 1 2 for TM mode
(8)
k2
k2n12
Now using Bloch’s theorem, E K (x , z ) = E K ( x )e −iβ z e − iKx , where EK(x+d) = EK(x) and simplifying we obtain the
dispersion relation for the proposed structure as given by

1
1
 1

K (ω ) =   cos −1  cos(k1a )cos (k 2 b ) −  γ +  sin (k1a ) sin (k 2 b )
2
γ
 d





(9)
Here, K is known as the Bloch wave number and is a function of ω as k1 and k 2 is the function of ω . For normal
incidence β = 0 , therefore, equation (9) is reduced to

1

1

1 1 n
n  
1 
1

K (ω ) =   cos−1 cos n1aω  cos n 2 bω  −  2 + 1  sin n1aω  sin  n 2bω 
c
c  2  n1 n 2  
c 
c
d



( )




(10)
As mentioned earlier, the group velocity Vg can be defined appropriately in the photonic crystals as usual [12],
−1
 dK 
(11)
Vg = 

 dω 
Now the expression for the effective group index of refraction (n eff (g )) can be calculated by using the formula,
n eff (g ) =
c
Vg
(12)
The effective refractive index neff(p) can be given as ,
n eff ( p) =
c.K (ω)
ω
(13)
The coefficient of reflection of the proposed structure can be deduced by using the relation [22]
b
R N =  0
 a0



b N =0
(14)
where a0 and b0 represents of the complex amplitudes of incident plane wave and reflected plane wave and the condition
b N = 0 implies the boundary condition that to the right of periodic structure there is no wave incident on it.
By solving this equation using N th power of a unimodular matrix, we obtain the expressions for the reflectivity of the
structure, as shown below
2
C
=
2
,
(15)
2
 sin Kd 
C +

 sin NKd 
where N is the number of unit cells (i.e. number of pairs of layers). The value of K can be determined from the equation
(9).
RN
2
3. RESULTS
In this section, we have computed the reflectivity of the PPC structure, numerically, for a given set of incident angles
and for different values of the plasma thickness in the far-ultraviolet and visible regions of the electromagnetic spectrum.
Furthermore, anomalous behavior of group velocity and negative effective refractive index has also been computed.
3.1 Reflectivity: Here, for the sake of numerical computation we take alternate layers of dielectric materials
( MgF2 , ZnS ) and plasma. The refractive index of dielectric is taken as n1 = 1.38(MgF 2) and 2.35(ZnS) and that of
1
  ω 2  2
13

p
  for which we chose ω p = 5.6 ×10 Hz and
the plasma is taken according to the expression n 2 = 1 − 



  ω  


L= 500 nm. L is the width of dielectric materials and Ld is that of plasma. For the proposed PPC structure we have
taken N= 15, and the angle of incidence is varied as θ = 00, 150, 300 and 450. Ratio of plasma to the dielectric width
(d) is taken as d=0.1 and d=1.0. Substituting all these values in equation (15) we obtain the reflectivity vs. frequency
graphs for the proposed structure, as depicted in the figure 2 and figure 3. From the study of these figures it is found
that forbidden gap increases due to the increase in refractive index of the dielectric material. Moreover, when the
incident wave angle of the wave is tilted from the normal these band edges are shifted towards higher frequency
since the wave traverses more in the materials.
3.2 Anomalous Group Velocity and Effective Index of Refraction: For the numerical calculation of group velocity
and effective index of refraction we have taken the same structure as in the previous section. Figure 4 and figure 5
present the plot of group velocity, effective group index and effective refractive index in the far ultraviolet and
visible regions. By examining the graphs it is found that group velocity becomes negative in a certain frequency
range viz.,
(i) When n=1.38 ( MgF2 ) and d=0.1, neff(g) becomes negative for the frequency range 1241-2429 tera Hz and for
d=1.0, it becomes negative for the frequency range 814-1489 tera Hz and 2378-2943 tera Hz .
(ii) When n=2.35 ( ZnS ) and d=0.1 negative neff(g) are observed at frequency range 783-1503 tera Hz and 24612943 tera Hz. For d=1.0 negative neff(g ) are observed between 643- 996 tera Hz.
It is interesting to note that, as the value of d increases, the range of negative group velocity becomes narrower and
at the same time minimum and maximum values of the group velocity change. This anomalous behavior of group
velocity inside the PPC structure causes the negative value of effective refractive index in the same frequency range
in which Vg becomes negative.
4. CONCLUSIONS
The band gap becomes larger with the increase of the plasma width. The dispersion relation for far ultraviolet-visible
spectrum shows that frequency gaps become larger for large plasma width and high refractive index of the dielectric
materials. The ranges of reflectivity also follow the same pattern. The range of 100% reflectivity increases with the
increase in the angle of incidence. It has been also observed that the structure with high refractive index contrast and
large plasma width yields large group velocity. For MgF 2-plasma structure, neff(g) becomes ultrahigh negative whereas
for the ZnS-plasma structure it becomes ultrahigh positive near the band edges. However, neff(p) does not show
anomalous behavior for both the structures considered in the study.
ACKNOLEDGEMENT
Mr. S. K. Singh acknowledges the Council of Scientific and Industrial Research (CSIR), INDIA for providing financial
assistance.
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*[email protected], [email protected],