JOURNAL OF TELECOMMUNICATIONS, VOLUME 18, ISSUE 1, JANUARY 2013
© 2012 JOT www.journaloftelecommunications.co.uk
13
Analytical Analysis of ray characteristics inside the optical fiber
*Chakresh Kumar,**Girish Narah and **Aroop Sharma
Abstract-In this paper we obtain solution of the ray equations in a parabolic and elliptical refractive index profile. We studied the various
conditions for the suitable propagation of ray inside the fiber. A comparison is also made between the ray propagating in elliptical and
parabolic refractive index profile.
Keywords :- Graded index fibre, Ray theory concept, refractive index profile
INTRODUCTION:- We consider a medium of varying refractive
index n=n(x). According to snell’s law
n1sinΦ1=n2sinΦ2=constant ,Φ1, Φ2, are the angle of
incidence at various interfaces If Θ1, Θ2, ........are the
corresponding angles that the ray makes with z- axis
Then n1cosθ1=n2cosθ2=n3cosθ3=constant (β) (a)
When the refractive index variation is continuous, the
thickness of each layer becomes infinitesimally small
and it form a continuous curve as shown in above
figure and it is taken form reference [1]
n(x)cosθ(x)=β(invariant of ray path)
1)()(
(dz) + (dx) = (ds)
22
222
dzdxdzds
or
Cosθ = (ds
dz)
-1 =dz
ds
We obtain
(ds
dz)=
1
cos θ x =
n x
β[from eqn (a)]
Substituting ds
dzin eqn (b)
(dx
dz)
2= 1
)(2
2
xn
(c)
Differentiating eqn(c) with respect to z
dz
dx
dx
(x)dn1=
dz
xd
dz
dx2
2
22
2
dx
xdn
dz
xd )(
2
1 2
22
2
(1)
The above equation is another form of ray equation
Ray Path In Parabolic Refractive Index
The parabolic refractive index is characterised by the
following refractive index distribution
])(21[)( 22
1
2
a
xnxn When | x|< a (core)
=2
2
2
1 ]21[ nn when |x|>a (cladding) (2)
Applying equation (2) in equation (1), we get
])(21[2
1 22
122
2
a
xn
dx
d
dz
xd
= )}({2
22
2
1 zxa
n
*Assistant Professor Electronics and communication department Tezpur (central) University,Tezpur,Assam India
** UG studentsElectronics and communication department Tezpur (central) University,Tezpur,Assam India
JOURNAL OF TELECOMMUNICATIONS, VOLUME 18, ISSUE 1, JANUARY 2013
© 2012 JOT www.journaloftelecommunications.co.uk
14
)(2
2
2
zxdz
xd , Where
1
2n
a
)(2
2
2
zxdz
xd =0
Therefore the general solution is given by
x(z)=AsinГz + BcosГz (3)
and similarly we can find
y(z)=CsinГz + Dcos Гz (4)
Where A, B,C & D are the Constants which can be
determined by the initial launching condition of the
ray.Now, let us consider that an extreme form of
skew rays is launched on the x-axis(at x= a’) in the y-z
plane(making angle θ’ with the z-axis). Thus , the
launching conditions on plane z=0 are
xz=0=a’ =>B=a’
And 𝑑𝑥
𝑑𝑧|z=0=0 =>A=0
Putting the value of A and B in equation (3), we get :-
Thus x(z)=a’cosГz (5)
And
Yz=0=0 =>D=0
And 𝑑𝑦
𝑑𝑧|z=0= tanθ’
=>C=𝑡𝑎𝑛𝜃 ’
Г
=>C=2
'tana
1n
If at the launching point n=n’ then β=n’cosθ’, so
C=2
'sin'
1n
an
Putting the value of C and D in equation (4)
zn
anzy
sin
2
'sin')(
1
Now if an
na
2''sin 1 , then
zazy sin')( (6)
Suppose a=50μm, a’=20μm,Δ=0.04, θ’=15˚ the
propagation of the ray is shown in fig (1)
Fig (1)
Keeping the other factor fixed and varying θ’ such as
θ’=45˚ and θ’=90˚, we observe the propagation in fig
(2) and fig(3) respectively
Fig (2)
Fig (3)
0
500
1000
1500
2000
-20
-10
0
10
20-20
-10
0
10
20
z in micrometers
Helical ray propagation
y in micrometers
x in m
icro
mete
rs
0
500
1000
1500
2000
-20
-10
0
10
20-20
-10
0
10
20
z in micrometers
Helical ray propagation
y in micrometers
x in m
icro
mete
rs
0
500
1000
1500
2000
-20
-10
0
10
20-20
-10
0
10
20
z in micrometers
Helical ray propagation
y in micrometers
x in m
icro
mete
rs
JOURNAL OF TELECOMMUNICATIONS, VOLUME 18, ISSUE 1, JANUARY 2013
© 2012 JOT www.journaloftelecommunications.co.uk
15
The number of helical turn’s increases as the angle is
varied from 0 ˚ to 90˚ that means the ray paths get
denser as the angle is increased.
Suppose a=10μm, a’=20μm, Δ=0.04, θ=60˚ the
propagation of the ray is shown in fig (4)
Fig (4)
Now keeping the other terms constant and varying
core radius a such that a=20μm and a=60μm, the
propagation is shown in fig(5) and fig(6)
Fig (5)
Fig (6)
The number of helical turn’s decreases as the core
radius is increased as observed from the graphs in fig
(4),(5),(6)
Now in fig (4), if we change the value of Δ such that
Δ=0.03 and Δ=0.07, the propagation will be shown in
fig (7) and fig (8)
Fig (7)
Fig (8)
0
500
1000
1500
2000
-20
-10
0
10
20-20
-10
0
10
20
z in micrometers
Helical ray propagation
y in micrometers
x in m
icro
mete
rs
0
500
1000
1500
2000
-20
-10
0
10
20-20
-10
0
10
20
z in micrometers
Helical ray propagation
y in micrometers
x in m
icro
mete
rs
0
500
1000
1500
2000
-20
-10
0
10
20-20
-10
0
10
20
z in micrometers
Helical ray propagation
y in micrometers
x in
mic
rom
eter
s
0
500
1000
1500
2000
-20
-10
0
10
20-20
-10
0
10
20
z in micrometers
Helical ray propagation
y in micrometers
x in m
icro
mete
rs
0
500
1000
1500
2000
-20
-10
0
10
20-20
-10
0
10
20
z in micrometers
Helical ray propagation
y in micrometers
x in m
icro
mete
rs
JOURNAL OF TELECOMMUNICATIONS, VOLUME 18, ISSUE 1, JANUARY 2013
© 2012 JOT www.journaloftelecommunications.co.uk
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The number of helical turn’s increases as the value of
Δ is increased as observed from fig (4),(7),(8)Suppose
a=50μm, a’=5μm, Δ=0.04, θ=60˚ the propagation will
be as shown in fig (9)
Fig (9)
Keeping the other terms constant and changing a’
such that a’=10μm and a’=60μm, the propagation is
shown in fig (10) and fig(11)
Fig (10)
Fig (11)
The radius of helical turns increases as the launching
point is increased, as observed from the figure
(9),(10),(11)
Ray Path In Elliptical Refractive Index
Now we will obtain the ray paths in an elliptical
index fibre characterized by the following refractive
9.5
Fig (12)
Here a=2b. For 4a=b the propagation is shown in fig
(13)
Fig (13)
As clear from fig(12) and fig(13) there is no fixed
shape for the ray path in elliptical refractive index.
When a =b the elliptical refractive index behaves as
parabolic refractive index as show in fig(14)
0
500
1000
1500
2000
-5
0
5-5
0
5
z in micrometers
Helical ray propagation
y in micrometers
x in m
icro
mete
rs
0
500
1000
1500
2000
-10
-5
0
5
10-10
-5
0
5
10
z in micrometers
Helical ray propagation
y in micrometers
x in m
icro
mete
rs
0
500
1000
1500
2000
-100
-50
0
50
100-60
-40
-20
0
20
40
60
z in micrometers
Helical ray propagation
y in micrometers
x in m
icro
mete
rs
0
500
1000
1500
2000
-10
-5
0
5
10-15
-10
-5
0
5
10
15
z in micrometers
ray propagation in elliptical index
y in micrometers
x in m
icro
mete
rs
0
500
1000
1500
2000
-50
0
50-15
-10
-5
0
5
10
15
z in micrometers
ray propagation in elliptical index
y in micrometers
x in m
icro
mete
rs
JOURNAL OF TELECOMMUNICATIONS, VOLUME 18, ISSUE 1, JANUARY 2013
© 2012 JOT www.journaloftelecommunications.co.uk
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Fig (14)
CONCLUSION
In this paper ray characteristics in parabolic and
elliptical index profile fiber has been studied at
different parameter.We observed that the ray graphs
in the parabolic refractive index by varying core
radius (a), angle (θ'), relative core-cladding index
difference (Δ) and launching point(a’) . We conclude
that-
Keeping θ', Δ, a' constant and
varying core radius (a) we found
that the no. of helical path decreases
with increase in core radius.
Keeping a, Δ, a' constant and
varying θ' from 0˚ to 180˚ we found
that the no. of helical path increases
when angle is increased from 0˚ to
90˚ after that the no. of helical path
decreases with the increase in angle
after 90˚ to 180˚. No of helical path is
maximum at 90˚.
Keeping a, a', θ' constant and
varying Δ we found that the no. of
helical path increases with increase
in Δ.
Keeping a, θ', Δ constant and
varying launching point (a') we
found that the radius of each helical
path increases with increase in the
launching point.
Increase in the no. of helical path and radius of the
helical path result in internal time delay i.e. time
required to send data through the optical fiber will
increase with increase in angle from 0˚ to 90˚,
increase in Δ and increase in launching point (a') and
since no. of helical path decreases with increase in
core radius, therefore to avoid time delay the value of
angle (θ'), Δ and a' should be as less as possible and
core radius should be high.
References
1. Ajoy Ghatak, K. Thyagarajan,(1999).
Introduction to Fiber Optics,Cambridge University
press.
2. Checcacci, P. F.,(1980). “Applicability of an
asymptotic numerical method to the
determination of whispering and bouncing
modes in elliptical fibers,” Journal of the Optical
Society of America vol. 70 no. 12.
3. Pierre Aschiéri,(2006). “Complex behavior of a
ray in a Gaussian index profile periodically
segmented waveguide ” J. Opt. A: Pure Appl.
Opt. vol. 8 pp. 386.
4. Hagen Renner, (1998). “Polarization
characteristics of optical waveguides with
separable symmetric refractive-index profiles,”
Journal of the Optical Society of America A vol.
15, no. 5.
5.S.K RaghuwanshiRay Paths In An Elliptic
Parabolic Refractive Index Profile FiberWorld
Journal of Science and Technology 2011, 1(8): 74-
78ISSN: 2231 – 2587
0
500
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-20
-10
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20-15
-10
-5
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z in micrometers
ray propagation in elliptical index
y in micrometers
x in m
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