DEPENDENCE OF CUTOFF VALUES ON
PITCH ANGLE OF CIRCULAR HELICALLY
CLADDED OPTICAL WAVEGUIDE AT ITS
CORE CLADDING INTERFACE Ajay Kumar Gautam
1, Amit Kumar Katariya
2, Dr. Vivekanand Mishra
3
[email protected], [email protected], [email protected] 1, 2
Dev Bhoomi Institute of Technology, Dehradun, Uttarakhand 3Sr. Member IEEE
3Sardar Vallabhbhai National Institute of Technology, Surat, Gujarat
Abstract- This article includes dispersion characteristics of
optical waveguide with helical windings, 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.
I. INTRODUCTION
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].
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.
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.
II. THEORETICAL 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 cross-section 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. 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.
III. 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.
1 1 0zE sin E cos (1)
2 20
zE sin E cos (2)
1 2 1 2 0z zE E cos E E sin (3)
1 2 1 2 0z zH H sin H H cos (4)
IV. MODAL EQUATIONS
2 ( )j j z j t
ZE CK ua e
(7)
2
j j z j t
zH DK ua e
(8)
Where, , , ,A B C D are arbitrary constants which are to be
evaluated from the boundary conditions. Also J ua
and ( )K wa are the Bessel functions.
For a guided mode, the propagation constant lies between two
limits 2 and 1 . If 2 2 1 1n k k k n k then a field
distribution is generated which will has an oscillatory behavior
in the core and a decaying behavior in the cladding. The
energy then is propagated along fiber without any loss. Where
2k
is free – space propagation constant. The transverse
field components can be obtained by using Maxwell’s standard
relations. So the electric and magnetic field components Eϕ and
Hϕ can be written as,
The expressions for Eϕ and Hϕ inside the core are, when (r < a)
1 2( ) '( )
j j z j tjE j AJ ua uBJ ua e
u a
(9)
1 12( ) '( )
j j z j tjH j BJ ua uAJ ua e
u a
(10)
The expressions for Eϕ and Hϕ inside the core are, when (r > a)
2 2( ) '( )
j j z j tjE j CK wa wDK wa e
w a
(11)
2 22( ) '( )
j j z j tjH j DK wa wCK wa e
w a
(12)
Now put these transverse field components equations into
boundary conditions, we get following four unknown equations
involving four unknown arbitrary constants
2( ) sin cos '( ) cos 0
jAJ ua BJ ua
u a u
(13)
2( ) sin cos '( ) cos 0
jCK wa DK wa
w a w
(14)
2
2
( ) cos sin '( ) sin
( ) cos sin '( ) sin 0
jAJ ua BJ ua
u a u
jCK wa DK wa
w a w
(15)
1
2
2
2
'( ) cos ( ) sin cos
'( ) cos ( ) sin cos 0
jAJ ua BJ ua
u u a
jCK wa DK wa
w w a
(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
1 2 3 4
1 2 3 40
1 2 3 4
1 2 3 4
A A A A
B B B B
C C C C
D D D D
(17)
where,
21 ( ) sin cos
2 '( ) cos
3 0
4 0
A J uau a
jA J ua
u
A
A
(18)
2
1 0
2 0
3 ( ) sin cos
4 '( ) cos
B
B
B K waw a
jB K wa
w
(19)
2
2
1 ( ) cos sin
2 '( ) sin
3 ( ) cos sin
4 '( ) sin
C J uau a
jC J ua
u
C K waw a
jC K wa
w
(20)
1
2
2
2
1 '( ) cos
2 ( ) sin cos
3 '( ) cos
4 ( ) sin cos
jD J ua
u
D J uau a
jD K wa
w
D K waw a
(21)
Evaluation of the above determinant yields the following Eigen
value equation for β. The determinant can be solve as
1 2 3 42 3 4 1 3 4 1 2 4 1 2 3
1 2 3 41 2 3 4 2 1 3 4 3 1 2 4 4 1 32 3
1 2 3 42 3 4 1 3 4 1 2 4 1 2 3
1 2 3 4
A A A AB B B B B B B B B B B B
B B B BA C C C A C C C A C C C A C C C
C C C CD D D D D D D D D D D D
D D D D
1 2 3 4 1 2 0 0
1 2 3 4 0 0 3 4
1 2 3 4 1 2 3 4
1 2 3 4 1 2 3 4
A A A A A A
B B B B B B
C C C C C C C C
D D D D D D D D
1 2 3 40 3 4 0 3 4
1 2 3 41 2 3 4 2 1 3 4
1 2 3 42 3 4 1 3 4
1 2 3 4
A A A AB B B B
B B B BA C C C A C C C
C C C CD D D D D D
D D D D
1 2 3 4
1 2 3 41 3 2 4 4 2 4( 2 3 3 2)
1 2 3 4
1 2 3 4
2 3 1 4 4 1 4( 1 3 3 1)
A A A A
B B B BA B C D C D B C D C D
C C C C
D D D D
A B C D C D B C D C D
1 2 3 4
1 2 3 41 3 2 4 1 3 4 2 1 4 2 3 1 4 3 2
1 2 3 4
1 2 3 4
2 3 1 4 2 3 4 1 2 4 1 3 2 4 3 1
A A A A
B B B BA B C D A B C D A B C D A B C D
C C C C
D D D D
A B C D A B C D A B C D A B C D
After eliminating unknown constants from equations (17),
(18), (19), (20) & (21), we get the following characteristic
equation.
1
2
2 2
2
2
2 2
2
2
( ) '( )sin cos cos
'( ) ( )
( ) '( )sin cos cos 0
'( ) ( )
kJ ua J uau
J ua u a u J ua
kK wa K waw
K wa w a w K wa
(22)
Equation (22) is standard characteristic equation, and is used
for model dispersion properties and model cutoff conditions.
V. RESULTS
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.
1
2
2 2
2
2
2 2
2
2
( ) '( )sin cos cos
'( ) ( )
( ) '( )sin cos cos 0
'( ) ( )
kJ ua J uau
J ua u a u J ua
kK wa K waw
K wa w a w K wa
(23)
2 2 2
2
2 2
1 2
( / )k nawb
V n n
(24)
2
2 2 2 2 2 2
1 2
2( ) ( )
aV u w a n n
(25)
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, 30
0, 45
0, 60
0 and 90
0.
0 2 4 6 8 10 12 140
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
V
b
Figure 2: Dispersion Curve of normalized propagation constant b as a function of V for a lower – order modes for pitch angle ψ = 00
0 2 4 6 8 10 12 140
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
V
b
Figure 3: Dispersion Curve of normalized propagation constant b as a function of V for a lower – order modes for pitch angle ψ = 300
0 2 4 6 8 10 12 140
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
V
b
Figure 4: Dispersion Curve of normalized propagation constant b as a function of V for a lower – order modes for pitch angle ψ = 450
0 2 4 6 8 10 12 140
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
V
b
Figure 5: Dispersion Curve of normalized propagation constant b as a function of V for a lower – order modes for pitch angle ψ = 600
0 2 4 6 8 10 12 140
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
V
b
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, 45
0 and 60
0 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.
We now come to table 1. we note particularly that the
dependence of the cutoff V – value (Vc) on ψ is such that as ψ
is increased there is a drastic fall in Vc at ψ =300 and then a
small increase as ψ goes from 300 to 60
0; then is a quick rise as
ψ changes from 600 to 90
0 (Figure 7).
TABLE I
CUTOFF VC VALUES FOR SOME LOWER ORDER MODES
ψ Vc
00 1.80 3.80 4.00 6.90 7.10 10.10 10.30 - -
300 0.05 1.70 1.80 3.70 3.90 7.00 7.10 10.20 10.30
450 0.40 1.70 1.80 3.65 3.70 7.00 7.20 10.20 10.30
600 0.30 1.50 1.80 3.70 3.90 7.00 7.20 10.20 10.30
900 1.90 3.80 5.40 7.00 8.60 10.20 11.80 - -
0 10 20 30 40 50 60 70 80 900
2
4
6
8
10
12
Angle in Degree
Vc
Figure 7: Dependence of cutoff values Vc on the pitch angle ψ
Thus the two most sensitive regions in respect of the influence
of helical pitch angle ψ on the cutoff values and the model
properties of waveguides are ranges from ψ = 00 to ψ = 30
0 and
ψ = 600 to ψ = 90
0 and these ranges of pitch angle expected to
have potential applications with ψ as a means for controlling
the model properties.
VI. 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, 45
0 and 60
0 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.
REFERENCES
[1] Ajay Kumar Gautam, "Analytical Study of Helically Cladded Optical Waveguide with Different Pitch Angles", International Journal of Advanced Engineering Technology, Vol. II, Issue I, pp. - 144-153, January - March, 2011.
[2] Ajay Kumar Gautam, "Dispersion & Cut-off Characteristics of Circular Helically Cladded Optical Fiber", International Journal of Advanced Engineering Technology, Vol. II, Issue III, pp. - 297-305, July-September, 2011.
[3] V.N. Mishra, Vivek Singh, B. Prasad, S. P. Ojha (2000). Optical Dispersion curves of two metal - clad lightguides having double convex lens core cross sections. Wiley, Microwave and Optical Technology Letters, Vol. 24, No. 4, 229-232.
[4] V.N. Mishra, V. Singh, B. Prasad, S. P. Ojha(2000). An Analytical investigation of dispersion characteristic of a lightguide with an annular core cross section bounded by two cardioids. Wiley, Microwave and Optical Technology Letters, Vol. 24, No. 4, 229-232.
[5] V. Singh, S. P. Ojha, B. Prasad, and L. K. Singh (1999). Optical and microwave Dispersion curves of an optical waveguide with a guiding region having a core cross section with a lunar shape. Optik 110, 267-270.
[6] V. Singh, S. P. Ojha, and L. K. Singh (1999). Model Behavior, cutoff condition, and dispersion characteristics of an optical waveguide with a core cross section bounded by two spirals. Microwave Optical Technology Letter, Vol. 21, 121-124.
[7] V. Singh, S. P. Ojha, and B. Prasad (1999). Weak guidance modal dispersion characteristics of an optical waveguide having core with sinusoidally varying gear shaped cross section. Microwave Optical Technology Letter, Vol. 22, 129-133.
[8] Ajay Kumar Gautam, "Dispersion Characteristic of Optical Waveguide with Helical Winding for Different Pitch Angle", National Conference on Electronics, Communication & Instrumentation, April, 2- 3, 2010.
[9] P. K. Choudhury, D. Kumar, and Z. Yusoff, F. A. Rahman (2009). An analytical investigation of four-layer dielectric optical fibers with au nano-coating - A comparison with three-layer optical fibers. PIER 90, 269 - 286.
[10] Keiser G. (2000). Optical Fiber Communications (3rd ed.). Singapore: McGraw-Hill.
[11] Ajay Kumar Gautam, "Effect of Helical Pitch Angles on Dispersion Characteristics of Circular Optical Waveguide Having Helical Windings on Core - Cladding Interface", International Journal of Advanced Engineering Technology, Vol. II, Issue I, pp. - 283-291, July-September, 2011.
[12] Kumar, D. and O. N. Singh II (2008). Towards the dispersion relations for dielectric optical fibers with helical windings under slow and fast wave considerations – a comparative analysis. PIER, Vol. 80, 409–420.
[13] Kumar, D. and O. N. Singh II (2002). An analytical study of the modal characteristics of annular step – index fiber of elliptical cross – section with two conducting helical windings on the two boundary surfaces between the guiding and non – guiding regions. Optik, Vol. 113, No. 5, 193-196.
[14] Singh, U. N., O. N. Singh II, P. Khastgir and K. K. Dey (1995). Dispersion characteristics of helically cladded step – index optical fiber analytical study. J. Opt. Soc. 1273-1278.
[15] M. P. S. Rao, Vivek Singh, B. Presad and S. P. Ojha (1999). Model characteristic and dispersion curves of hypocycloidal optical waveguide. Optik, 110, No. 2, 81-85.
[16] Ajoy Ghatak and K. Thyagarajan (2008). Optical Electronics. India: Cambridge University Press.
[17] Kumar, D. and O. N. Singh II (2001). Some special cases of propagation characteristics of an elliptical step – index fiber with a conducting helical winding on the core – cladding boundary – An analytical treatment. Optik Vol. 112, No. 12, 561-566.
[18] Govind P. Agrawal (2002). Fiber – Optic Communication System (3rd
ed.). New York: A John Wiley & Sons, Inc. Publication.
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