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Helical Windings on Circular Optical Waveguide and Induced
Effects on its Dispersion Curve
Ajay Kumar Gautam
Amit Kumar Katariya
Izhar Ahmed
Departmant of Electronics &
Communication Engineering
Dev Bhoomi Institute of
Technology
Dehradun, India
Departmant of Electronics &
Communication Engineering
Dev Bhoomi Institute of
Technology
Dehradun, India
Departmant of Electronics &
Communication Engineering
Dev Bhoomi Institute of
Technology
Dehradun, India
[email protected]
[email protected]
[email protected]
ABSTRACT
This article includes dispersion characteristics of optical
waveguide with helical winding, and compression of
dispersion characteristics of optical waveguide with helical
winding at core-cladding interface for five different pitch
angles. In this article dispersion characteristic of conventional
optical waveguide with helical winding at core – cladding
interface has been obtained. The model dispersion
characteristics of optical waveguide with helical winding at
core-cladding interface have been obtained for five different
pitch angles. Boundary conditions have been used to obtain
the dispersion characteristics and these conditions have been
utilized to get the model Eigen values equation. From these
Eigen value equations dispersion curve are obtained and
plotted for modified optical waveguide for particular values of
the pitch angle of the winding and the effect of this winding
has been discussed. The article also shows the effect in the
Dispersion Curve with changing the Pitch Angle.
The conventional optical fiber having a circular core cross –
section which is widely used in optical communication
systems [1]. Recently metal – clad optical waveguides have
been studied because these provide potential applications,
connecting the optical components to other circuits. Metallic –
cladding structure on an optical waveguide is known as a TE
– mode pass polarizer and is commercially applied to various
optical devices [4]. The propagation characteristics of optical
fibers with elliptic cross – section have been investigated by
many researchers. Singh [5] have proposed an analytical study
of dispersion characteristics of helically cladded step – index
optical fiber with circular core. The model characteristic and
dispersion curves of a hypocycloidal optical waveguide have
been investigated by Ojha [6]. Present work is the study of
circular optical waveguide with sheath helix [3] in between
the core and cladding region. The sheath helix is a cylindrical
surface with high conductivity in a preferential direction
which winds helically at constant angle around the core –
cladding boundary surfaces.
Keywords
Bessel Functions, Dispersion Curves, Characteristics
Equation, Sheath Helix, Circular Waveguide, Modal Cutoff
INTRODUCTION
We An optical waveguide is basically a cylindrical dielectric
waveguide with a circular cross section where a high-index
wave guiding core is surrounded by a low-index cladding. The
index step and profile are controlled by the concentration and
distribution of dopants. Silica fibers are ideal for light
transmission in the visible and near-infrared regions because
of their low loss and low dispersion in these spectral regions.
They are therefore suitable for optical communications. Even
though optical fiber seems quite flexible, it is made of glass,
which cannot withstand sharp bending or longitudinal stress.
Therefore when fiber is placed inside complete cables special
construction techniques are employed to allow the fiber to
move freely within a tube. Usually fiber optic cables contain
several fibers, a strong central strength member and one or
more metal sheaths for mechanical protection. Some cables
also include copper pairs for auxiliary applications. Optical
fibers with helical winding are known as complex optical
waveguides. The use of helical winding in optical fibers
makes the analysis much accurate. As the number of
propagating modes depends on the helix pitch angle, so
helical winding at core – cladding interface can control the
dispersion characteristics of the optical waveguide [3].
Optical fibers with helical winding are known as complex
optical waveguides. The conventional optical fiber having a
circular core cross – section which is widely used in optical
communication systems. The use of helical winding in optical
fibers makes the analysis much accurate [1]. The propagation
characteristics of optical fibers with elliptic cross – section
have been investigated by many researchers. Singh [13] have
proposed an analytical study of dispersion characteristics of
helically cladded step – index optical fiber with elliptical core.
Present work is the study of circular optical waveguide with
sheath helix in between the core and cladding region, this
work also gives the comparison of dispersion characteristic at
different pitch angles. The sheath helix [12] is a cylindrical
surface with high conductivity in a preferential direction
which winds helically at constant angle around the core –
cladding boundary surfaces. As the number of propagating
modes depends on the helix pitch angle [2], so helical winding
at core-cladding interface can control the dispersion
characteristics [3-7] of the optical waveguide. The winding
angle of helix (ψ) can take any arbitrary value between 0 to
π/2. In case of sheath helix winding [1], cylindrical surface
with high conductivity in the direction of winding which
winds helically at constant pitch angle (ψ) around the core
cladding boundary surface. We assume that the waveguide
have real constant refractive index of core and cladding is n1
and n2 respectively (n1 > n2). In this type of optical wave
guide which we get after winding, the pitch angle controls the
model characteristics of optical waveguide..
Theoritical Background
The optical waveguide is the fundamental element that
interconnects the various devices of an optical integrated
circuit, just as a metallic strip does in an electrical integrated
circuit. However, unlike electrical current that flows through a
metal strip according to Ohm’s law, optical waves travel in
the waveguide in distinct optical modes. A mode, in this
sense, is a spatial distribution of optical energy in one or more
dimensions that remains constant in time. The mode theory,
along with the ray theory, is used to describe the propagation
of light along an optical fiber. The mode theory [10] is used to
describe the properties of light that ray theory is unable to
explain. The mode theory uses electromagnetic wave behavior
to describe the propagation of light along a fiber. A set of
guided electromagnetic waves is called the modes [13, 16] of
the fiber. For a given mode, a change in wavelength can
prevent the mode from propagating along the fiber. The mode
is no longer bound to the fiber. The mode is said to be cut off
[13]. Modes that are bound at one wavelength may not exist at
longer wavelengths. The wavelength at which a mode ceases
to be bound is called the cutoff wavelength [11] for that
mode. However, an optical fiber is always able to propagate at
least one mode. This mode is referred to as the fundamental
mode [16] of the fiber. The fundamental mode can never be
cut off. We can take a case of a fiber with circular crosssection wound with a sheath helix at the core-clad interface
(Figure 1). A sheath helix can be assumed by winding a very
thin conducting wire around the cylindrical surface so that the
spacing between the nearest two windings is very small and
yet they are insulated from each another. In our structure, the
helical windings are made at a constant helix pitch angle (ψ).
We assume that (n1-n2) / n1 << 1.
Waveguide with Conducting Helical
Winding
We consider the case of a fiber with circular cross – section
wrapped with a sheath helix at core – clad boundary as shown
in Figure 1.
Figure 1: Fiber with circular cross – section wrapped with a sheath helix
In our structure, the helical windings are made at a constant
angle ψ – the helix pitch angle. The structure has high
conductivity in a preferential direction. The pitch angle can
control the propagation behavior of such fibers [23]. We
assume that the core and cladding regions have the real
refractive indices n1 and n2 (n1 > n2), and (n1-n2) / n1 << 1. The
winding is right – handed and the direction of propagation is
positive z direction. The winding angle of the helix (pitch
angle - ψ) can take any arbitrary value between 0 to π/2. This
type of fibers is referred to as circular helically cladded fiber
(CHCF). This analysis requires the use of cylindrical
coordinate system ( r ,  , z ) [18] with the z – axis being the
direction of propagation.
Boundary Conditions
Tangential component of the electric field in the direction of
the conducting winding should be zero, and in the direction
perpendicular to the helical winding, the tangential component
of both the electric field and magnetic field must be
continuous, so we have following boundary condition [17]
with helix.
Ez1sin  E1cos  0
(1)
Ez 2 sin  E 2cos  0
(2)
 Ez1  Ez 2  cos   E1  E 2  sin
0
(3)
 H z1  H z 2  sin   H1  H 2  cos
0
(4)
conditions must be satisfied n2 k  k2    k1  n1k , where
n1 and n2 are refractive indices or core and cladding regions
respectively. The solution of the axial field components can
be written as,
Modal Equation
The guided mode along this type of fiber can be analyzed in a
standard way, with the cylindrical coordinates system
( r ,  , z ) . In order to have a guided field the following
The expressions for Ez and Hz inside the core are, when (r < a)
Ez1  AJ (ua)e j  j z  jt
H z1  BJ  ua  e
(5)
j  j  z  jt
(6)
The
expressions for Ez and Hz outside the core are, when (r > a)
EZ 2  CK (ua)e j  j  z  jt
H z 2  DK  ua  e
where,
A, B, C , D
are arbitrary constants which are to be
evaluated from the boundary conditions. Also
K ( wa)
J  ua  and
are the Bessel functions.
For a guided mode, the propagation constant lies between two
limits  2 and 1 . If
n2 k  k2    k1  n1k
then a
(7)
j  j  z  jt
(8)
behavior in the core and a decaying behavior in the cladding.
The energy then is propagated along fiber without any loss.
Where
k
2

is free – space propagation constant. The
transverse field components can be obtained by using
Maxwell’s standard relations.
field distribution is generated which will has an oscillatory
The expressions for Eϕ and Hϕ inside the core are, when (r < a)
j  

j
AJ (ua)  uBJ '(ua)  e j  j  z  jt
2 
u  a

j  

H1   2  j
BJ (ua )  1uAJ '(ua )  e j  j  z  jt
u  a

E1  
(9)
(10)
The expressions for Eϕ and Hϕ inside the core are, when (r > a)
j  

j
CK ( wa )   wDK '( wa )  e j  j  z  jt
w2  a

j  

 2 j
DK ( wa )   2 wCK '( wa )  e j  j  z  jt
w  a

E 2  
H 2
(11)
(12)
Now put these transverse field components equations into boundary conditions, we get following four unknown equations involving
four unknown arbitrary constants



 j  
AJ (ua) sin  2 cos   BJ '(ua) 
cos  0
u a


 u 



 j  
CK ( wa ) sin  2 cos   DK '( wa ) 
cos  0
wa


 w 
(13)
(14)



 j  
AJ (ua ) cos  2 sin   BJ '(ua) 
sin
u a


 u 



 j  
CK ( wa) cos  2 sin   DK '( wa) 
sin  0
wa


 w 

 j1 


 AJ '(ua ) 
cos  BJ (ua ) sin  2 cos 

u a


 u 

 j 2 


CK '( wa ) 
cos  DK ( wa ) sin  2 cos   0

wa


 w 
(15)
(16)
Equations (13), (14), (15) and (16) will yield a non – trivial solution if the determinant whose elements are the coefficient of these
unknown constants is set equal to zero. Thus we have
A1 A2 A3 A4
B1 B 2 B3 B 4
0
C1C 2 C 3 C 4
(17)
D1 D 2 D3 D 4
where,



A1  J (ua) sin  2 cos 
ua


 j  
A2  J '(ua) 
cos
 u 
A3  0
A4  0
B1  0
B2  0



B3  K ( wa) sin  2 cos 
wa


 j  
B 4  K '( wa) 
cos
 w 



C1  J (ua) cos  2 sin 
u a


 j  
C 2   J '(ua) 
sin
 u 



C 3   K ( wa) cos  2 sin 
wa


 j  
C 4  K '( wa) 
sin
 w 
(18)
(19)
(20)
 j1 
D1   J '(ua) 
cos
 u 



D 2  J (ua) sin  2 cos 
u
a


 j 2 
D3  K '( wa ) 
cos
 w 
(21)



D 4   K ( wa) sin  2 cos 
wa


After eliminating unknown constants from equations (17), (18), (19), (20) & (21), we get the following characteristic equation.
2
2
J (ua) 

 k1 J '(ua )
u
cos 2 
 sin  2 cos  
J '(ua ) 
u a
u J (ua )

2
k 2 2 K '( wa )
K ( wa ) 


w
cos 2   0
 sin  2 cos  
K '( wa) 
wa
w K ( wa )

(22)
Equation (22) is standard characteristic equation, and is used for model dispersion properties and model cutoff conditions.
Simulation Results and Discussion
It is now possible to interpret the characteristic equation (Equation 22) in numerical terms. This will give us an insight into model
properties of our waveguide.
2
2
J (ua) 

 k1 J '(ua )
u
cos 2 
 sin  2 cos  
J '(ua ) 
u a
u J (ua )

2
k 2 2 K '( wa )
K ( wa ) 


w
cos 2   0
 sin  2 cos  
K '( wa) 
wa
w K ( wa )

(23)
2
2
 aw   (  / k )  n2 
b

 
2
2
 V   n1  n2

(24)
 2 a 
2
2
V 2  (u 2  w2 )a 2  
 (n1  n2 )
  
(25)
2
2
where, b & V are known as normalization propagation
constant & normalized frequency parameter respectively. We
make some simple calculations based on Equations (24) and
(25). We choose n1=1.50, n2=1.46 and λ =1.55µm. We take
  1 for simplicity, but the result is valid for any value of
.
In order to plot the dispersion relations, we plot the
normalized frequency parameter V against the normalization
propagation constant b. we considered five special cases
corresponding to the values of pitch angle ψ as 00, 300, 450,
600 and 900.
1
0.9
0.8
0.7
b
0.6
0.5
0.4
0.3
0.2
0.1
0
0
2
4
6
8
10
12
14
V
Figure 2: Dispersion Curve of normalized propagation constant b as a function of V for a lower – order modes for pitch angle
ψ = 00
1
0.9
0.8
0.7
b
0.6
0.5
0.4
0.3
0.2
0.1
0
0
2
4
6
8
10
12
14
V
Figure 3: Dispersion Curve of normalized propagation constant b as a function of V for a lower – order modes for pitch angle
ψ = 300
1
0.9
0.8
0.7
b
0.6
0.5
0.4
0.3
0.2
0.1
0
0
2
4
6
8
10
12
14
V
Figure 4: Dispersion Curve of normalized propagation constant b as a function of V for a lower – order modes for pitch angle
ψ = 450
1
0.9
0.8
0.7
b
0.6
0.5
0.4
0.3
0.2
0.1
0
0
2
4
6
8
10
12
14
V
Figure 5: Dispersion Curve of normalized propagation constant b as a function of V for a lower – order modes for pitch angle
ψ = 600
1
0.9
0.8
0.7
b
0.6
0.5
0.4
0.3
0.2
0.1
0
0
2
4
6
8
10
12
14
V
Figure 6: Dispersion Curve of normalized propagation constant b as a function of V for a lower – order modes for pitch angle
ψ = 900
From the above figures we observe that, they all have
standard expected shape, but except for lower order modes
they comes in pairs, that is cutoff values for two adjacent
mode converge. This means that one effect of conducting
helical winding is to split the modes and remove a degeneracy
which is hidden in conventional waveguide without windings.
We also observe that another effect of the conducting helical
winding is to reduce the cutoff values, thus increasing the
number of modes. This effect is undesirable for the possible
use of these waveguide for long distance communication.
An anomalous feature in the dispersion curves is observable
for ψ = 300, 450 and 600 for this type of waveguide near the
lowest order mode. It is found that on the left of the lowest
cutoff values, portions of curves appear which have no
resemblance with standard dispersion curves, and have no
cutoff values. This means that for very small value of V
anomalous dispersion properties may occur in helically
wound waveguides.
We found that some curves have band gaps of discontinuities
between some value of V. These represent the band gaps or
forbidden bands of the structure. These are induced by the
periodicity of the helical windings.
Conclusion
From the above results we observe that, the effect of the
conducting helical winding is to reduce the cutoff values, thus
increasing the number of modes. This effect is undesirable for
the possible use of these waveguide for long distance
communication.
We also observe that, all curves have standard expected
shape, but except for lower order modes they comes in pairs,
that is cutoff values for two adjacent modes converge. This
means that one effect of conducting helical winding is to split
the modes and remove a degeneracy which is hidden in
conventional waveguide without windings.
An anomalous feature in the dispersion curves is observable
for ψ = 300, 450 and 600 for this type of waveguide near the
lowest order mode. It is found that on the left of the lowest
cutoff values, portions of curves appear which have no
resemblance with standard dispersion curves, and have no
cutoff values. This means that for very small value of V
anomalous dispersion properties may occur in helically
wound waveguides.
We found that some curves have band gaps of discontinuities
between some value of V. These represent the band gaps or
forbidden bands of the structure. These are induced by the
periodicity of the helical windings. Thus helical pitch angle
controls the modal properties of this type of optical
waveguide.
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